Construct ordinal sums of triangular norms on pairwise non-overlapped subintervals in meet semi-lattices 在交会半格中对成对不重叠的子区间构造三角范数的序数和
Hongliang Lai *, Yang Luo 赖洪亮*, 罗扬School of Mathematics, Sichuan University, Chengdu 610064, China 四川大学数学学院，中国成都 610064
Received 6 July 2022; received in revised form 26 October 2022; accepted 28 October 2022 收到日期：2022 年 7 月 6 日；修订后收到日期：2022 年 10 月 26 日；接受日期：2022 年 10 月 28 日
Available online 3 November 2022 2022 年 11 月 3 日在线可用
To construct semigroups from existing ones, the ordinal sum construction is introduced in [3,15]. Particularly, for t -norms, a kind of special ordered semigroups on the unit interval [0,1][0,1], ordinal sums play a fundamental role. It is well known that every continuous t-norm on [0,1][0,1] is an ordinal sum of a family of continuous Archimedean t-norms [1,13][1,13]. 为了从现有的半群构造新的半群，在文献[3,15]中引入了序和构造。特别是对于 t-范数，在单位区间 [0,1][0,1] 上，一种特殊的有序半群中，序和起着基础性作用。众所周知，单位区间 [0,1][0,1] 上的每一个连续 t-范数都是一族连续阿基米德 t-范数的序和 [1,13][1,13] 。
In recent years, ordered semigroups on bounded lattices, or more generally, on bounded posets, with the top element being the unit, are investigated also under the name tt-norms [2,4-7,10,12,16-21][2,4-7,10,12,16-21]. Saminger constructs the ordinal sum of t -norms on a bounded lattice [19,20] by extending the ordinal sums on the unit interval [0,1] straightforwardly. Let LL be a bounded meet semi-lattice and II be a totally ordered index set. Further, let ]a_(i),b_(i)[∣i in I}:}\left] a_{i}, b_{i}[\mid i \in I\}\right. be a family of pairwise disjoint subintervals of LL and each T_(i)T_{i} be a tt-norm on the interval [a_(i),b_(i)]\left[a_{i}, b_{i}\right]. Then the ordinal sum TT is given by 近年来，受限格上的有序半群，或更一般地说，在有界偏序集上，顶元素为单位的情况，也被称为 tt -范数 [2,4-7,10,12,16-21][2,4-7,10,12,16-21] 。Saminger 通过直接扩展单位区间[0,1]上的序数和，构造了受限格上的 t-范数的序数和[19,20]。设 LL 为一个有界交半格， II 为一个全序索引集。此外，设 ]a_(i),b_(i)[∣i in I}:}\left] a_{i}, b_{i}[\mid i \in I\}\right. 为 LL 的成对不相交子区间的家族，每个 T_(i)T_{i} 为区间 [a_(i),b_(i)]\left[a_{i}, b_{i}\right] 上的 tt -范数。那么序数和 TT 由以下公式给出：
T(x,y)={[T_(i)(x","y)",",x","y in[a_(i),b_(i)]],[x^^y","," otherwise "]:}T(x, y)= \begin{cases}T_{i}(x, y), & x, y \in\left[a_{i}, b_{i}\right] \\ x \wedge y, & \text { otherwise }\end{cases}
However, as shown in [19], the binary operation TT may fail to be associative because in a lattice there may be incomparable elements with the endpoints of the involved subintervals. Accordingly, Saminger’s ordinal sum of t - 然而，如[19]所示，二元运算 TT 可能不满足结合律，因为在一个格中，可能存在与所涉及子区间的端点不可比较的元素。因此，Saminger 的序数和 t -
norms does not always yield a t-norm even if only one summand is involved [14,19]. Therefore, it is not trivial to construct a t-norm which is the ordinal sum of a family of tt-norms on the intervals in a bounded poset. 规范并不总是产生 t-规范，即使只涉及一个加数 [14,19]。因此，构造一个在有界偏序集的区间上是一组 tt -规范的序和的 t-规范并不是微不足道的。
Roughly speaking, the aim of constructing the ordinal sum of the family T_(i)T_{i} is to find a t-norm defined globally on the poset LL which coincides with each T_(i)T_{i} on [a_(i),b_(i)]\left[a_{i}, b_{i}\right] locally. When the index set II is finite and the endpoints of the intervals form a chain, it is shown that the t-norm defined globally on the bounded meet semi-lattice LL does always exist [8]. Moreover, for the index set Z\mathbb{Z} consisting of all integral numbers, if the endpoints of all intervals [a_(i),b_(i)]\left[a_{i}, b_{i}\right] form a chain such that b_(i) <= a_(i+1)b_{i} \leq a_{i+1} for all i inZi \in \mathbb{Z}, it is shown that the global t-norm on LL always exists whenever LL is a complete lattice [18]. Generally, the obtained t-norms on LL are different from the binary operation given by the Eq. (1). 粗略地说，构造家族 T_(i)T_{i} 的序和的目的是在偏序集 LL 上找到一个全局定义的 t-范数，该范数在局部上与每个 T_(i)T_{i} 在 [a_(i),b_(i)]\left[a_{i}, b_{i}\right] 上重合。当索引集 II 是有限的，并且区间的端点形成一个链时，已证明在有界交半格 LL 上全局定义的 t-范数总是存在[8]。此外，对于由所有整数构成的索引集 Z\mathbb{Z} ，如果所有区间的端点 [a_(i),b_(i)]\left[a_{i}, b_{i}\right] 形成一个链，使得对于所有 i inZi \in \mathbb{Z} 都有 b_(i) <= a_(i+1)b_{i} \leq a_{i+1} ，则已证明在 LL 上的全局 t-范数总是存在，只要 LL 是一个完备格[18]。一般来说，在 LL 上获得的 t-范数与方程(1)给出的二元运算是不同的。
In this paper, for any given totally ordered index set II, our strategy is to collect all elements comparable with all the endpoints of the intervals [a_(i),b_(i)],i in I\left[a_{i}, b_{i}\right], i \in I. These elements form a subset bar(C)sube L\bar{C} \subseteq L, which contains each subinterval [a_(i),b_(i)],i in I\left[a_{i}, b_{i}\right], i \in I. Furthermore, the binary operation TT given by Eq. (1) is a t-norm on bar(C)\bar{C} indeed and coincides with each T_(i)T_{i} on [a_(i),b_(i)]\left[a_{i}, b_{i}\right] locally. Moreover, by aiding of the right adjoint of the inclusion map bar(C)↪L\bar{C} \hookrightarrow L (if it exists), one can extend the t-norm on bar(C)\bar{C} to the whole bounded meet semi-lattice LL. 在本文中，对于任何给定的全序索引集 II ，我们的策略是收集与区间 [a_(i),b_(i)],i in I\left[a_{i}, b_{i}\right], i \in I 的所有端点可比的所有元素。这些元素形成一个子集 bar(C)sube L\bar{C} \subseteq L ，其中包含每个子区间 [a_(i),b_(i)],i in I\left[a_{i}, b_{i}\right], i \in I 。此外，由公式(1)给出的二元运算 TT 确实是 bar(C)\bar{C} 上的一个 t-范数，并且在 [a_(i),b_(i)]\left[a_{i}, b_{i}\right] 上与每个 T_(i)T_{i} 局部一致。此外，通过辅助包含映射 bar(C)↪L\bar{C} \hookrightarrow L 的右伴随（如果存在），可以将 t-范数从 bar(C)\bar{C} 扩展到整个有界交半格 LL 。
The content of this paper is organized as follows. In Section 2, we recall some basic notions of t-norms on bounded posets. In Section 3, we construct the ordinal sums of a family of triangular norms on a bounded meet semi-lattice, which are our main results. In Section 4, as applications of the main results, we include some examples of ordinal sums appearing in the literature. 本文的内容组织如下。在第二节中，我们回顾了有界偏序集上的 t-范数的一些基本概念。在第三节中，我们构造了有界交半格上三角范数族的序和，这是我们的主要结果。在第四节中，作为主要结果的应用，我们包括了一些文献中出现的序和的例子。
2. Preliminaries 2. 初步准备
In this section, we recall some basic notions of t-norms on bounded lattices. 在本节中，我们回顾有界格上的 t-范数的一些基本概念。
A poset PP is a nonempty set PP equipped with an order relation <=\leq (i.e., a reflexive, antisymmetric and transitive binary relation). A poset PP is called a totally ordered set, or a chain, if for all a,b in Pa, b \in P, it holds that either a <= ba \leq b or b <= ab \leq a. The principal ideals of a poset PP are the subsets darr x={y in P∣y <= x}\downarrow x=\{y \in P \mid y \leq x\}. The greatest lower bound of two element xx and yy is called the meet of xx and yy, denoted by x^^yx \wedge y. Dually, the least upper bound of two element of xx and yy is called the join of xx and yy, denoted by x vv yx \vee y. A poset LL is called a meet semi-lattice if any two elements xx and yy have a meet, and is called a lattice if any two elements xx and yy have both the meet and the join. For a,b in La, b \in L, we write a < ba<b for that a <= ba \leq b but a!=ba \neq b. 一个偏序集 PP 是一个非空集合 PP ，配备有一个顺序关系 <=\leq （即，一个自反、反对称和传递的二元关系）。如果对于所有 a,b in Pa, b \in P ，要么 a <= ba \leq b ，要么 b <= ab \leq a ，则称偏序集 PP 为全序集或链。偏序集 PP 的主理想是子集 darr x={y in P∣y <= x}\downarrow x=\{y \in P \mid y \leq x\} 。两个元素 xx 和 yy 的最大下界称为 xx 和 yy 的交，记作 x^^yx \wedge y 。对偶地，两个元素 xx 和 yy 的最小上界称为 xx 和 yy 的并，记作 x vv yx \vee y 。如果任何两个元素 xx 和 yy 都有交，则称偏序集 LL 为交半格；如果任何两个元素 xx 和 yy 都有交和并，则称为格。对于 a,b in La, b \in L ，我们写 a < ba<b 表示 a <= ba \leq b 但 a!=ba \neq b 。
A poset PP is bounded if it has a top element, written as 1 , and a bottom element, written as 0 . Throughout this paper, unless otherwise stated, we always assume that LL is a bounded meet semi-lattice with the top element 1 and the bottom element 0 . 一个偏序集 PP 是有界的，如果它有一个上界元素，记作 1 ，以及一个下界元素，记作 0 。在本文中，除非另有说明，我们始终假设 LL 是一个有界的交半格，具有上界元素 1 和下界元素 0 。
Let LL be a meet semi-lattice, a,b in La, b \in L and a <= ba \leq b. The subinterval [a,b][a, b] is defined as 让 LL 成为一个交会半格， a,b in La, b \in L 和 a <= ba \leq b 。子区间 [a,b][a, b] 定义为
[a,b]=:{x in L∣a <= x <= b}[a, b]=:\{x \in L \mid a \leq x \leq b\}
and ]a,b[] a, b[ is defined as 和 ]a,b[] a, b[ 被定义为
]a,b[=:{x in L∣a < x < b}.] a, b[=:\{x \in L \mid a<x<b\} .
Two subintervals [a,b][a, b] and [c,d][c, d] are said to be non-overlapped if ]a,b[nn]c,d[=O/] a, b[\cap] c, d[=\emptyset. 两个子区间 [a,b][a, b] 和 [c,d][c, d] 被称为不重叠，如果 ]a,b[nn]c,d[=O/] a, b[\cap] c, d[=\emptyset 。
Obviously, [a,b][a, b] is a bounded meet semi-lattice with the top element bb and the bottom element aa. 显然， [a,b][a, b] 是一个有界的交半格，顶元素是 bb ，底元素是 aa 。
Definition 2.1 ([6]). Let PP be a bounded poset. A binary operation T:P^(2)longrightarrow PT: P^{2} \longrightarrow P is called a triangular norm (or a t-norm for short) on PP, if for all x,y,z in Px, y, z \in P, it satisfies the following four properties: 定义 2.1 ([6])。设 PP 为一个有界偏序集。二元运算 T:P^(2)longrightarrow PT: P^{2} \longrightarrow P 称为三角范数（或简称 t-范数），如果对于所有 x,y,z in Px, y, z \in P ，它满足以下四个性质：
(1) T(x,y)=T(y,x)T(x, y)=T(y, x) (commutativity); (1) T(x,y)=T(y,x)T(x, y)=T(y, x) （交换律）；
(2) T(x,y) <= T(x,z)T(x, y) \leq T(x, z), if y <= zy \leq z (increasingness); (2) T(x,y) <= T(x,z)T(x, y) \leq T(x, z) ，如果 y <= zy \leq z （递增性）；
(3) T(x,T(y,z))=T(T(x,y),z)T(x, T(y, z))=T(T(x, y), z) (associativity); (3) T(x,T(y,z))=T(T(x,y),z)T(x, T(y, z))=T(T(x, y), z) （结合性）；
(4) T(x,1)=T(1,x)=xT(x, 1)=T(1, x)=x (neutrality). (4) T(x,1)=T(1,x)=xT(x, 1)=T(1, x)=x （中立性）。
For a bounded poset PP, there is a t -norm, called the drastic product t -norm: 对于一个有界的偏序集 PP ，存在一种 t-范数，称为剧烈乘积 t-范数：
T_(D)(x,y)={[x^^y," if "1in{x","y}],[0," otherwise "]:}T_{D}(x, y)= \begin{cases}x \wedge y & \text { if } 1 \in\{x, y\} \\ 0 & \text { otherwise }\end{cases}
The drastic product T_(D)T_{D} is the smallest t-norm on PP under the pointwise order. If PP is a bounded meet semi-lattice, the binary operation ^^\wedge becomes a t-norm, denoted by T_(M)T_{M}. In this case, T_(M)T_{M} is the largest t -norm on PP under the pointwise order. 剧烈乘积 T_(D)T_{D} 是在点序下对 PP 的最小 t-范数。如果 PP 是一个有界交半格，则二元运算 ^^\wedge 变成一个 t-范数，记作 T_(M)T_{M} 。在这种情况下， T_(M)T_{M} 是在点序下对 PP 的最大 t-范数。
In the fuzzy set theory, the unit interval [0,1][0,1] consisting of real numbers plays the fundamental role among the bounded posets. The t-norm T_(M)T_{M} on the unit interval [0,1] is often called the Gödel t-norm. There are another two basic continuous t-norms T_(P)T_{P} and T_(E)T_{\mathrm{E}} on the unit interval: 在模糊集理论中，由实数组成的单位区间 [0,1][0,1] 在有界偏序集中发挥着基础作用。单位区间[0,1]上的 t-范数 T_(M)T_{M} 通常被称为哥德尔 t-范数。单位区间上还有另外两个基本的连续 t-范数 T_(P)T_{P} 和 T_(E)T_{\mathrm{E}} ：
the product t-norm: T_(P)(x,y)=x*yT_{P}(x, y)=x \cdot y. 产品 t-norm: T_(P)(x,y)=x*yT_{P}(x, y)=x \cdot y 。
Given a family of pairwise disjoint open intervals ]a_(i),e_(i)[∣i in I}:}\left] a_{i}, e_{i}[\mid i \in I\}\right. in the unit interval [0,1][0,1] and for all i in Ii \in I, there is a t-norm T_(i)T_{i} on [ a_(i),e_(i)a_{i}, e_{i} ], one can constitute a t-norm TT as follows [13]. 给定单位区间 [0,1][0,1] 中的一组成对不相交的开区间 ]a_(i),e_(i)[∣i in I}:}\left] a_{i}, e_{i}[\mid i \in I\}\right. ，对于所有 i in Ii \in I ，在[ a_(i),e_(i)a_{i}, e_{i} ]上存在一个 t-范数 T_(i)T_{i} ，可以如下构造 t-范数 TT [13]。
T(x,y)={[T_(i)(x","y)," if "(x","y)in[a_(i),e_(i)]^(2)],[x^^y," otherwise. "]:}T(x, y)= \begin{cases}T_{i}(x, y) & \text { if }(x, y) \in\left[a_{i}, e_{i}\right]^{2} \\ x \wedge y & \text { otherwise. }\end{cases}
The binary operation TT is called the ordinal sums of the family {T_(i):i in I}\left\{T_{i}: i \in I\right\}. 二元运算 TT 称为家族 {T_(i):i in I}\left\{T_{i}: i \in I\right\} 的序数和。
In this paper, we focus on the question: given a meet semi-lattice LL and a family of pairwise non-overlapped intervals {[a_(i),b_(i)]∣i in I}\left\{\left[a_{i}, b_{i}\right] \mid i \in I\right\}, where II is a chain such that b_(i) <= a_(j)b_{i} \leq a_{j} whenever i <= ji \leq j, and a t -norm T_(i)T_{i} on each interval [a_(i),b_(i)]\left[a_{i}, b_{i}\right], can one obtain an ordinal sum of the family {T_(i)∣i in I}\left\{T_{i} \mid i \in I\right\} ? 在本文中，我们关注的问题是：给定一个满足半格 LL 和一组成对不重叠的区间 {[a_(i),b_(i)]∣i in I}\left\{\left[a_{i}, b_{i}\right] \mid i \in I\right\} ，其中 II 是一个链，当 b_(i) <= a_(j)b_{i} \leq a_{j} 时 i <= ji \leq j ，以及在每个区间 [a_(i),b_(i)]\left[a_{i}, b_{i}\right] 上的 t-范数 T_(i)T_{i} ，是否可以获得该家族 {T_(i)∣i in I}\left\{T_{i} \mid i \in I\right\} 的序数和？
3. Main results 3. 主要结果
In this section, we always assume that LL is a bounded meet semi-lattice, II is a totally ordered index set and {[a_(i),b_(i)]∣i in I}\left\{\left[a_{i}, b_{i}\right] \mid i \in I\right\} is a family of intervals in LL such that b_(i) <= a_(j)b_{i} \leq a_{j} whenever i <= ji \leq j, that is, the family {[a_(i),b_(i)]∣i in I}\left\{\left[a_{i}, b_{i}\right] \mid i \in I\right\} is pairwise non-overlapped. 在本节中，我们始终假设 LL 是一个有界的交会半格， II 是一个全序的索引集， {[a_(i),b_(i)]∣i in I}\left\{\left[a_{i}, b_{i}\right] \mid i \in I\right\} 是 LL 中的一组区间，使得 b_(i) <= a_(j)b_{i} \leq a_{j} 每当 i <= ji \leq j 时，即该家族 {[a_(i),b_(i)]∣i in I}\left\{\left[a_{i}, b_{i}\right] \mid i \in I\right\} 是成对不重叠的。
Clearly, the endpoints of all intervals [a_(i),b_(i)]\left[a_{i}, b_{i}\right] form a chain in LL, which is denoted by 显然，所有区间 [a_(i),b_(i)]\left[a_{i}, b_{i}\right] 的端点在 LL 中形成一个链，记作
C=:{a_(i)∣i in I}uu{b_(i)∣i in I}C=:\left\{a_{i} \mid i \in I\right\} \cup\left\{b_{i} \mid i \in I\right\}
Collect all the elements in LL which are comparable with all the elements in CC, one obtains a subset of LL, denoted by 收集所有在 LL 中与 CC 中所有元素可比较的元素，可以得到 LL 的一个子集，记作
bar(C)=:{x in L∣" for all "c in C," either "x <= c" or "c <= x}\bar{C}=:\{x \in L \mid \text { for all } c \in C, \text { either } x \leq c \text { or } c \leq x\}
We claim that bar(C)\bar{C} satisfies the following properties: 我们声称 bar(C)\bar{C} 满足以下属性：
Both 0 and 1 are in bar(C)\bar{C}, and for all i in I,[a_(i),b_(i)]sube bar(C)i \in I,\left[a_{i}, b_{i}\right] \subseteq \bar{C}. 0 和 1 都在 bar(C)\bar{C} 中，并且对于所有 i in I,[a_(i),b_(i)]sube bar(C)i \in I,\left[a_{i}, b_{i}\right] \subseteq \bar{C} 。
bar(C)=L\bar{C}=L whenever LL is totally ordered. bar(C)=L\bar{C}=L 每当 LL 完全有序时。
bar(C)\bar{C} is closed with respect to the infima and suprema existing in LL. Therefore, bar(C)\bar{C} is a meet sub-semi-lattice of LL. Moreover, if LL is a complete lattice, then bar(C)\bar{C} is also a complete sub-lattice of LL. In fact, given a family {x_(j)∣j in:}\left\{x_{j} \mid j \in\right.J}sube bar(C)J\} \subseteq \bar{C} with xx being the supremum in LL, one has that x <= c Longleftrightarrow AA j in J,x_(j) <= cx \leq c \Longleftrightarrow \forall j \in J, x_{j} \leq c. Hence, if x≰cx \not \leq c, there is some j in Jj \in J such that x_(j)≰cx_{j} \not \leq c, that is, c <= x_(j)c \leq x_{j} since x_(j)in bar(C)x_{j} \in \bar{C}. It follows that c <= x_(j) <= xc \leq x_{j} \leq x and x in bar(C)x \in \bar{C} holds as desired. One can check the infimum i n f_(j in J)x_(j)\inf _{j \in J} x_{j} is also in bar(C)\bar{C} similarly. bar(C)\bar{C} 在 LL 中关于下确界和上确界是闭合的。因此， bar(C)\bar{C} 是 LL 的一个交半格。此外，如果 LL 是一个完备格，那么 bar(C)\bar{C} 也是 LL 的一个完备子格。实际上，给定一个家族 {x_(j)∣j in:}\left\{x_{j} \mid j \in\right.J}sube bar(C)J\} \subseteq \bar{C} ，其中 xx 是 LL 中的上确界，可以得出 x <= c Longleftrightarrow AA j in J,x_(j) <= cx \leq c \Longleftrightarrow \forall j \in J, x_{j} \leq c 。因此，如果 x≰cx \not \leq c ，则存在某个 j in Jj \in J 使得 x_(j)≰cx_{j} \not \leq c ，即 c <= x_(j)c \leq x_{j} ，因为 x_(j)in bar(C)x_{j} \in \bar{C} 。因此， c <= x_(j) <= xc \leq x_{j} \leq x 和 x in bar(C)x \in \bar{C} 按所需成立。可以类似地检查下确界 i n f_(j in J)x_(j)\inf _{j \in J} x_{j} 也在 bar(C)\bar{C} 中。
The theorem in the below is a consequence of Proposition 5.2 in [19], but we check it here straightforwardly for the convenience of readers. 下面的定理是[19]中命题 5.2 的结果，但我们在这里直接检查它，以方便读者。
Theorem 3.1. Let T_(i)T_{i} be a tt-norm on [a_(i),b_(i)]\left[a_{i}, b_{i}\right] for all i in Ii \in I, then the ordinal sum T={(:[a_(i),b_(i)],T_(i):)}_(i in I): bar(C)^(2)longrightarrow bar(C)T=\left\{\left\langle\left[a_{i}, b_{i}\right], T_{i}\right\rangle\right\}_{i \in I}: \bar{C}^{2} \longrightarrow \bar{C} given by 定理 3.1. 设 T_(i)T_{i} 是对所有 i in Ii \in I 的 [a_(i),b_(i)]\left[a_{i}, b_{i}\right] 上的 tt -范数，则由以下给出的序数和 T={(:[a_(i),b_(i)],T_(i):)}_(i in I): bar(C)^(2)longrightarrow bar(C)T=\left\{\left\langle\left[a_{i}, b_{i}\right], T_{i}\right\rangle\right\}_{i \in I}: \bar{C}^{2} \longrightarrow \bar{C}
bar(T)(x,y)={[T_(i)(x","y)," if "x","y in[a_(i),b_(i)]],[x^^y," otherwise "]:}\bar{T}(x, y)= \begin{cases}T_{i}(x, y) & \text { if } x, y \in\left[a_{i}, b_{i}\right] \\ x \wedge y & \text { otherwise }\end{cases}
is a tt-norm on bar(C)\bar{C}. 是一个 tt -范数在 bar(C)\bar{C} 上。
Proof. Firstly, one can easily see that bar(T)\bar{T} has the unit element 1 and it is commutative. 证明。首先，可以很容易地看到 bar(T)\bar{T} 具有单位元素 1，并且它是可交换的。
Secondly, we show that bar(T)\bar{T} is associative, that is, bar(T)( bar(T)(x,y),z)= bar(T)(x, bar(T)(y,z))\bar{T}(\bar{T}(x, y), z)=\bar{T}(x, \bar{T}(y, z)) for all x,yx, y and zz in bar(C)\bar{C}. We check it in the following cases: 其次，我们证明 bar(T)\bar{T} 是结合的，即对于所有 x,yx, y 和 zz 在 bar(C)\bar{C} 中都有 bar(T)( bar(T)(x,y),z)= bar(T)(x, bar(T)(y,z))\bar{T}(\bar{T}(x, y), z)=\bar{T}(x, \bar{T}(y, z)) 。我们在以下情况下进行检查：
(1) There is an interval [a_(i),b_(i)]\left[a_{i}, b_{i}\right] such that x,y,z in[a_(i),b_(i)]x, y, z \in\left[a_{i}, b_{i}\right]. The associativity of bar(T)\bar{T} immediately follows from the associativity of T_(i)T_{i}. (1) 存在一个区间 [a_(i),b_(i)]\left[a_{i}, b_{i}\right] 使得 x,y,z in[a_(i),b_(i)]x, y, z \in\left[a_{i}, b_{i}\right] 。 bar(T)\bar{T} 的结合性直接来自于 T_(i)T_{i} 的结合性。
(2) There is an interval [a_(i),b_(i)]\left[a_{i}, b_{i}\right] including two elements, say xx and yy, but b_(i) < zb_{i}<z. One has that bar(T)( bar(T)(x,y),z)=\bar{T}(\bar{T}(x, y), z)=bar(T)(T_(i)(x,y),z)=T_(i)(x,y)^^z=T_(i)(x,y)= bar(T)(x,y)= bar(T)(x,y^^z)= bar(T)(x, bar(T)(y,z))\bar{T}\left(T_{i}(x, y), z\right)=T_{i}(x, y) \wedge z=T_{i}(x, y)=\bar{T}(x, y)=\bar{T}(x, y \wedge z)=\bar{T}(x, \bar{T}(y, z)). （2）有一个区间 [a_(i),b_(i)]\left[a_{i}, b_{i}\right] ，包括两个元素，即 xx 和 yy ，但 b_(i) < zb_{i}<z 。有一个 bar(T)( bar(T)(x,y),z)=\bar{T}(\bar{T}(x, y), z)=bar(T)(T_(i)(x,y),z)=T_(i)(x,y)^^z=T_(i)(x,y)= bar(T)(x,y)= bar(T)(x,y^^z)= bar(T)(x, bar(T)(y,z))\bar{T}\left(T_{i}(x, y), z\right)=T_{i}(x, y) \wedge z=T_{i}(x, y)=\bar{T}(x, y)=\bar{T}(x, y \wedge z)=\bar{T}(x, \bar{T}(y, z)) 。
(3) There is an interval [a_(i),b_(i)]\left[a_{i}, b_{i}\right] including two elements, say xx and yy, but z < a_(i)z<a_{i}. One has that bar(T)( bar(T)(x,y),z)=\bar{T}(\bar{T}(x, y), z)=bar(T)(T_(i)(x,y),z)=T_(i)(x,y)^^z=z=x^^z= bar(T)(x,z)= bar(T)(x,y^^z)= bar(T)(x, bar(T)(y,z))\bar{T}\left(T_{i}(x, y), z\right)=T_{i}(x, y) \wedge z=z=x \wedge z=\bar{T}(x, z)=\bar{T}(x, y \wedge z)=\bar{T}(x, \bar{T}(y, z)). （3）存在一个区间 [a_(i),b_(i)]\left[a_{i}, b_{i}\right] ，包括两个元素，即 xx 和 yy ，但 z < a_(i)z<a_{i} 。有一个条件是 bar(T)( bar(T)(x,y),z)=\bar{T}(\bar{T}(x, y), z)=bar(T)(T_(i)(x,y),z)=T_(i)(x,y)^^z=z=x^^z= bar(T)(x,z)= bar(T)(x,y^^z)= bar(T)(x, bar(T)(y,z))\bar{T}\left(T_{i}(x, y), z\right)=T_{i}(x, y) \wedge z=z=x \wedge z=\bar{T}(x, z)=\bar{T}(x, y \wedge z)=\bar{T}(x, \bar{T}(y, z)) 。
(4) There is no interval [a_(i),b_(i)]\left[a_{i}, b_{i}\right] including any two of x,yx, y and zz. The associativity of bar(T)\bar{T} immediately follows from the associativity of the meet ^^\wedge. （4）没有间隔 [a_(i),b_(i)]\left[a_{i}, b_{i}\right] ，包括 x,yx, y 和 zz 中的任何两个。 bar(T)\bar{T} 的结合性直接来自于交集 ^^\wedge 的结合性。
The proof of the associativity of bar(T)\bar{T} is finished. bar(T)\bar{T} 的结合性证明已完成。
Thirdly, we show that bar(T)\bar{T} is increasing. Take x,y,z in bar(C)x, y, z \in \bar{C} such that y <= zy \leq z, and consider the following cases: 第三，我们证明 bar(T)\bar{T} 是递增的。取 x,y,z in bar(C)x, y, z \in \bar{C} 使得 y <= zy \leq z ，并考虑以下几种情况：
(1) All elements x,yx, y and zz lie in some interval [a_(i),b_(i)]\left[a_{i}, b_{i}\right]. The increasingness of bar(T)\bar{T} immediately follows from the increasingness of T_(i)T_{i}. (1) 所有元素 x,yx, y 和 zz 都位于某个区间 [a_(i),b_(i)]\left[a_{i}, b_{i}\right] 内。 bar(T)\bar{T} 的递增性直接源于 T_(i)T_{i} 的递增性。
(2) The elements xx and yy lie in some interval [a_(i),b_(i)]\left[a_{i}, b_{i}\right] but b_(i) < zb_{i}<z. One has that bar(T)(x,y)=T_(i)(x,y) <= x=x^^z=\bar{T}(x, y)=T_{i}(x, y) \leq x=x \wedge z=bar(T)(x,z)\bar{T}(x, z). (2) 元素 xx 和 yy 位于某个区间 [a_(i),b_(i)]\left[a_{i}, b_{i}\right] 但 b_(i) < zb_{i}<z 。有 bar(T)(x,y)=T_(i)(x,y) <= x=x^^z=\bar{T}(x, y)=T_{i}(x, y) \leq x=x \wedge z=bar(T)(x,z)\bar{T}(x, z) 。
(3) The elements xx and zz lie in some interval [a_(i),b_(i)]\left[a_{i}, b_{i}\right] but y < a_(i)y<a_{i}. One has that bar(T)(x,y)=x^^y=y < a_(i) <= bar(T)(x,z)\bar{T}(x, y)=x \wedge y=y<a_{i} \leq \bar{T}(x, z). (3) 元素 xx 和 zz 位于某个区间 [a_(i),b_(i)]\left[a_{i}, b_{i}\right] 但 y < a_(i)y<a_{i} 。有 bar(T)(x,y)=x^^y=y < a_(i) <= bar(T)(x,z)\bar{T}(x, y)=x \wedge y=y<a_{i} \leq \bar{T}(x, z) 。
(4) Neither yy nor zz lies in same interval [a_(i),b_(i)]\left[a_{i}, b_{i}\right] with xx. One has that bar(T)(x,y)=x^^y <= x^^z= bar(T)(x,z)\bar{T}(x, y)=x \wedge y \leq x \wedge z=\bar{T}(x, z). (4) yy 和 zz 都不在与 xx 相同的区间 [a_(i),b_(i)]\left[a_{i}, b_{i}\right] 内。可以得出 bar(T)(x,y)=x^^y <= x^^z= bar(T)(x,z)\bar{T}(x, y)=x \wedge y \leq x \wedge z=\bar{T}(x, z) 。
Thus, the binary operation bar(T)\bar{T} is increasing as desired. 因此，二元运算 bar(T)\bar{T} 是按预期增加的。
Therefore, the ordinal sum bar(T)\bar{T} is indeed a t-norm on bar(C)\bar{C}. 因此，序数和 bar(T)\bar{T} 确实是 bar(C)\bar{C} 上的 t-范数。
To extend the ordinal sum t-norms bar(T)\bar{T} on bar(C)\bar{C} to the bounded meet semi-lattice LL, we need recall the notions on Galois connections. 为了将序数和 t-范数 bar(T)\bar{T} 从 bar(C)\bar{C} 扩展到有界交半格 LL ，我们需要回顾 Galois 连接的概念。
Let f:P longrightarrow Qf: P \longrightarrow Q and g:Q longrightarrow Pg: Q \longrightarrow P be a pair of maps between posets. We say that ff is left adjoint to gg, and gg is right adjoint to ff, if 让 f:P longrightarrow Qf: P \longrightarrow Q 和 g:Q longrightarrow Pg: Q \longrightarrow P 成为偏序集之间的一对映射。我们说 ff 是 gg 的左伴随，而 gg 是 ff 的右伴随，如果
f(x) <= y Longleftrightarrow x <= g(y)f(x) \leq y \Longleftrightarrow x \leq g(y)
for all x in Px \in P and y in Qy \in Q. In this case, we say that the pair (f,g)(f, g) is a Galois connection (or an adjunction), written as f⊣gf \dashv g. We say that ff is the left adjoint and gg is the right adjoint. 对于所有 x in Px \in P 和 y in Qy \in Q 。在这种情况下，我们说对偶 (f,g)(f, g) 是一个伽罗瓦连接（或附加），记作 f⊣gf \dashv g 。我们说 ff 是左伴随， gg 是右伴随。
Some basic properties of Galois connection are collected in the following proposition. 一些伽罗瓦连接的基本性质汇集在以下命题中。
Proposition 3.2 ([11]). Let f⊣g:P longrightarrow Qf \dashv g: P \longrightarrow Q be a Galois connection between posets PP and QQ, then the following properties hold. 命题 3.2 ([11])。设 f⊣g:P longrightarrow Qf \dashv g: P \longrightarrow Q 是偏序集 PP 和 QQ 之间的伽罗瓦连接，则以下性质成立。
(1) x <= gf(x)x \leq g f(x) and fg(y) <= yf g(y) \leq y for all x in Px \in P and y in Qy \in Q. (1) x <= gf(x)x \leq g f(x) 和 fg(y) <= yf g(y) \leq y 对于所有 x in Px \in P 和 y in Qy \in Q 。
(2) f(x)=fgf(x),g(y)=gfg(y)f(x)=f g f(x), g(y)=g f g(y) for all x in Px \in P and y in Qy \in Q. (2) f(x)=fgf(x),g(y)=gfg(y)f(x)=f g f(x), g(y)=g f g(y) 对于所有 x in Px \in P 和 y in Qy \in Q 。
Theorem 3.3 ([21]). Let PP and QQ be bounded posets, and g:P longrightarrow Q,d:Q longrightarrow Pg: P \longrightarrow Q, d: Q \longrightarrow P are order-preserving functions such that (d,g)(d, g) is a Galois connection. If TT is a tt-norm on QQ such that for all x,y in P,T(g(x),g(y))in g(P)uu{omega inx, y \in P, T(g(x), g(y)) \in g(P) \cup\{\omega \inQ∣omega <= g(0)}Q \mid \omega \leq g(0)\}, then the following binary operation T_(g):P^(2)longrightarrow PT_{g}: P^{2} \longrightarrow P is a tt-norm: 定理 3.3 ([21])。设 PP 和 QQ 为有界偏序集， g:P longrightarrow Q,d:Q longrightarrow Pg: P \longrightarrow Q, d: Q \longrightarrow P 为保持顺序的函数，使得 (d,g)(d, g) 是一个 Galois 连接。如果 TT 是 tt -范数，定义在 QQ 上，使得对于所有 x,y in P,T(g(x),g(y))in g(P)uu{omega inx, y \in P, T(g(x), g(y)) \in g(P) \cup\{\omega \inQ∣omega <= g(0)}Q \mid \omega \leq g(0)\} ，则以下二元运算 T_(g):P^(2)longrightarrow PT_{g}: P^{2} \longrightarrow P 是一个 tt -范数：
T_(g)(x,y)={[dT(g(x)","g(y)),x!=1" and "y!=1],[x^^y,x=1" or "y=1]:}T_{g}(x, y)= \begin{cases}d T(g(x), g(y)) & x \neq 1 \text { and } y \neq 1 \\ x \wedge y & x=1 \text { or } y=1\end{cases}
Recall that an order preserving map int: P longrightarrow PP \longrightarrow P is an interior operator on the poset PP if for all x in Px \in P, 回想一下，如果对于所有 x in Px \in P ，映射 int: P longrightarrow PP \longrightarrow P 是偏序集 PP 上的一个内部运算符，则它是一个保序映射
int x <= x,quad int(int x)=int x\operatorname{int} x \leq x, \quad \operatorname{int}(\operatorname{int} x)=\operatorname{int} x
Let int: P longrightarrow PP \longrightarrow P be an interior operator, then the inclusion map i:int(P)↪Pi: \operatorname{int}(P) \hookrightarrow P has a right adjoint g:P longrightarrow int(P)g: P \longrightarrow \operatorname{int}(P) given by g(x)=int(x)g(x)=\operatorname{int}(x) for all x in Px \in P. Conversely, given an inclusion map i:Q↪Pi: Q \hookrightarrow P with a right adjoint g:P longrightarrow Qg: P \longrightarrow Q, one can obtain an interior operator 让内点算子 P longrightarrow PP \longrightarrow P 为一个内部算子，则包含映射 i:int(P)↪Pi: \operatorname{int}(P) \hookrightarrow P 具有右伴随 g:P longrightarrow int(P)g: P \longrightarrow \operatorname{int}(P) ，其由 g(x)=int(x)g(x)=\operatorname{int}(x) 给出，对于所有 x in Px \in P 。反之，给定一个具有右伴随 g:P longrightarrow Qg: P \longrightarrow Q 的包含映射 i:Q↪Pi: Q \hookrightarrow P ，可以得到一个内部算子。
int:P longrightarrow P,int x=g(i(x))=g(x)\operatorname{int}: P \longrightarrow P, \operatorname{int} x=g(i(x))=g(x)
The following corollary generalizes Theorem 4.2 in [18] slightly. 以下推论稍微概括了[18]中的定理 4.2。
Corollary 3.4. Let LL be a bounded poset, M sube PM \subseteq P be the range of an interior operator int : P longrightarrow PP \longrightarrow P and VV be aa t-norm on MM. Then VV can be extended to PP via int, i.e., the binary operation TT on PP given by 推论 3.4. 设 LL 为一个有界偏序集， M sube PM \subseteq P 为内运算符 int : P longrightarrow PP \longrightarrow P 的范围， VV 为 aa 在 MM 上的 t-范数。那么 VV 可以通过 int 扩展到 PP ，即在 PP 上给出的二元运算 TT 。
T(x,y)={[V(int(x)","int(y)),x!=1" and "y!=1],[x^^y,x=1" or "y=1]:}T(x, y)= \begin{cases}V(\operatorname{int}(x), \operatorname{int}(y)) & x \neq 1 \text { and } y \neq 1 \\ x \wedge y & x=1 \text { or } y=1\end{cases}
is a tt-norm on PP. 是一个 tt -范数在 PP 上。
The inclusion map i: bar(C)↪Li: \bar{C} \hookrightarrow L does have right adjoints in a wide sense, as shown in the below. 包含映射 i: bar(C)↪Li: \bar{C} \hookrightarrow L 在广义上确实具有右伴随，如下所示。
Theorem 3.5. Let LL be a bounded meet semi-lattice. Then the inclusion map i: bar(C)↪Li: \bar{C} \hookrightarrow L has a right adjoint g:L longrightarrowg: L \longrightarrowbar(C)\bar{C} iffor all x in Lx \in L, the set 定理 3.5。设 LL 为一个有界交半格。那么包含映射 i: bar(C)↪Li: \bar{C} \hookrightarrow L 具有右伴随 g:L longrightarrowg: L \longrightarrowbar(C)\bar{C} ，如果对于所有 x in Lx \in L ，集合
M_(x)={x}uu{c in C∣c≰x}M_{x}=\{x\} \cup\{c \in C \mid c \not \leq x\}
has an infimum in L. In this case, g(x)=i n fM_(x)g(x)=\inf M_{x}. Moreover, if LL is a bounded lattice, the condition is also necessary. 在 L 中有一个下确界。在这种情况下， g(x)=i n fM_(x)g(x)=\inf M_{x} 。此外，如果 LL 是一个有界格，则该条件也是必要的。
Proof. Let g(x)g(x) be the infimum of the set M_(x)M_{x}. We show that g:L longrightarrow bar(C)g: L \longrightarrow \bar{C} is a right adjoint of the inclusion map i: bar(C)↪Li: \bar{C} \hookrightarrow L, that is, for all x in Lx \in L and y in bar(C),y <= x Longleftrightarrow y <= g(x)y \in \bar{C}, y \leq x \Longleftrightarrow y \leq g(x). Since g(x)g(x) is a lower bound of M_(x)M_{x}, one has that g(x) <= xg(x) \leq x. If y <= g(x)y \leq g(x), then y <= g(x) <= xy \leq g(x) \leq x. Conversely, if y <= xy \leq x, then y <= cy \leq c for all c in Cc \in C with c≰xc \not \leq x, otherwise, c < yc<y implies that c < y <= xc<y \leq x, which is a contradiction. Therefore, yy is a lower bound of M_(x)M_{x} and y <= g(x)y \leq g(x) as desired. 证明。设 g(x)g(x) 为集合 M_(x)M_{x} 的下确界。我们证明 g:L longrightarrow bar(C)g: L \longrightarrow \bar{C} 是包含映射 i: bar(C)↪Li: \bar{C} \hookrightarrow L 的右伴随，即对于所有 x in Lx \in L 和 y in bar(C),y <= x Longleftrightarrow y <= g(x)y \in \bar{C}, y \leq x \Longleftrightarrow y \leq g(x) 。由于 g(x)g(x) 是 M_(x)M_{x} 的下界，因此有 g(x) <= xg(x) \leq x 。如果 y <= g(x)y \leq g(x) ，则 y <= g(x) <= xy \leq g(x) \leq x 。反之，如果 y <= xy \leq x ，则对于所有 c in Cc \in C 且 c≰xc \not \leq x ，有 y <= cy \leq c ，否则， c < yc<y 意味着 c < y <= xc<y \leq x ，这是一种矛盾。因此， yy 是 M_(x)M_{x} 和 y <= g(x)y \leq g(x) 的下界，如所愿。
Suppose that LL is a bounded lattice and the inclusion map i: bar(C)↪Li: \bar{C} \hookrightarrow L has a right adjoint g:L longrightarrow bar(C)g: L \longrightarrow \bar{C}, that is, for all y in bar(C)y \in \bar{C} and x in L,y <= g(x)Longleftrightarrow y <= xx \in L, y \leq g(x) \Longleftrightarrow y \leq x. We check that g(x)g(x) is the infimum of M_(x)M_{x} in LL. Firstly, since g(x) <= g(x)g(x) \leq g(x), one has that g(x) <= xg(x) \leq x. Secondly, g(x) <= cg(x) \leq c for all c in Cc \in C with cxc \not x, otherwise, one has that c < g(x) <= xc<g(x) \leq x since c in Cc \in C, which is a contradiction. Thus, g(x)g(x) is a lower bound of M_(x)M_{x}. Thirdly, for any y in Ly \in L which is a lower bound of M_(x)M_{x}, y vv g(x)y \vee g(x) is also a lower bound of M_(x)M_{x}. Notice that for all c in C,c <= xc \in C, c \leq x implies that c <= g(x) <= g(x)vv yc \leq g(x) \leq g(x) \vee y. Hence, one has that y vv g(x)y \vee g(x) is also in bar(C)\bar{C}. So, one obtains that g(x)vv y <= g(x)g(x) \vee y \leq g(x) because y vv g(x) <= xy \vee g(x) \leq x and gg is a right adjoint of the inclusion map. Therefore, one has that y <= y vv g(x) <= g(x)y \leq y \vee g(x) \leq g(x), which implies that g(x)g(x) is the infimum of M_(x)M_{x} in LL. 假设 LL 是一个有界格，包含映射 i: bar(C)↪Li: \bar{C} \hookrightarrow L 有一个右伴随 g:L longrightarrow bar(C)g: L \longrightarrow \bar{C} ，即对于所有 y in bar(C)y \in \bar{C} 和 x in L,y <= g(x)Longleftrightarrow y <= xx \in L, y \leq g(x) \Longleftrightarrow y \leq x 。我们检查 g(x)g(x) 是 M_(x)M_{x} 在 LL 中的下确界。首先，由于 g(x) <= g(x)g(x) \leq g(x) ，可以得出 g(x) <= xg(x) \leq x 。其次，对于所有 g(x) <= cg(x) \leq c ，当 cxc \not x 时， c in Cc \in C ，否则，由于 c in Cc \in C ，可以得出 c < g(x) <= xc<g(x) \leq x ，这与事实相矛盾。因此， g(x)g(x) 是 M_(x)M_{x} 的下界。第三，对于任何 y in Ly \in L ，它是 M_(x)M_{x} 的下界， y vv g(x)y \vee g(x) 也是 M_(x)M_{x} 的下界。注意，对于所有 c in C,c <= xc \in C, c \leq x ，这意味着 c <= g(x) <= g(x)vv yc \leq g(x) \leq g(x) \vee y 。因此，可以得出 y vv g(x)y \vee g(x) 也在 bar(C)\bar{C} 中。所以，可以得出 g(x)vv y <= g(x)g(x) \vee y \leq g(x) ，因为 y vv g(x) <= xy \vee g(x) \leq x 和 gg 是包含映射的右伴随。因此，可以得出 y <= y vv g(x) <= g(x)y \leq y \vee g(x) \leq g(x) ，这意味着 g(x)g(x) 是 M_(x)M_{x} 在 LL 中的下确界。
Corollary 3.6. Let LL be a complete lattice, that is, any subset of LL has an infimum, then the inclusion map i: bar(C)↪Li: \bar{C} \hookrightarrow L has a right adjoint. 推论 3.6. 设 LL 是一个完备格，即 LL 的任何子集都有下确界，则包含映射 i: bar(C)↪Li: \bar{C} \hookrightarrow L 具有右伴随。
Corollary 3.7. Let LL be a bounded meet semi-lattice and CC be a well-ordered chain in L. Then for each x in Lx \in L, the set {c in C∣c≰x}\{c \in C \mid c \not \leq x\} has an minimum c_(x)c_{x} and x^^c_(x)x \wedge c_{x} is the infimum of M_(x)M_{x}. Therefore, the inclusion map i: bar(C)↪Li: \bar{C} \hookrightarrow L has a right adjoint. 推论 3.7. 设 LL 为一个有界交半格， CC 为 L 中的一个良序链。那么对于每个 x in Lx \in L ，集合 {c in C∣c≰x}\{c \in C \mid c \not \leq x\} 有一个最小元素 c_(x)c_{x} ，并且 x^^c_(x)x \wedge c_{x} 是 M_(x)M_{x} 的下确界。因此，包含映射 i: bar(C)↪Li: \bar{C} \hookrightarrow L 有一个右伴随。
Generally, the inclusion map i: bar(C)↪Li: \bar{C} \hookrightarrow L may fail to have a right adjoint as shown in the below. 一般来说，包含映射 i: bar(C)↪Li: \bar{C} \hookrightarrow L 可能没有右伴随，如下所示。
Example 3.8. Let L={0}xx[-1,0[uu({0,1}xx]0,1])L=\{0\} \times[-1,0[\cup(\{0,1\} \times] 0,1]), which is a subset of the plane R^(2)\mathbb{R}^{2}. For all points (x,y)(x, y) and (x^('),y^('))\left(x^{\prime}, y^{\prime}\right) in LL, define the partial order <=\leq by 例 3.8. 设 L={0}xx[-1,0[uu({0,1}xx]0,1])L=\{0\} \times[-1,0[\cup(\{0,1\} \times] 0,1]) 为平面 R^(2)\mathbb{R}^{2} 的一个子集。对于 LL 中的所有点 (x,y)(x, y) 和 (x^('),y^('))\left(x^{\prime}, y^{\prime}\right) ，通过以下方式定义偏序 <=\leq ：
(x,y) <= (x^('),y^('))Longleftrightarrow x <= x^(')" and "y <= y^(')(x, y) \leq\left(x^{\prime}, y^{\prime}\right) \Longleftrightarrow x \leq x^{\prime} \text { and } y \leq y^{\prime}
then LL becomes a bounded lattice. Given a chain 然后 LL 变成一个有界格。给定一个链
However, the inclusion i: bar(C)↪Li: \bar{C} \hookrightarrow L has no right adjoint. In fact, take a point (0,1)(0,1) in LL, then one can see that 然而，包含 i: bar(C)↪Li: \bar{C} \hookrightarrow L 没有右伴随。实际上，取 LL 中的一个点 (0,1)(0,1) ，那么可以看出
The lower bounds of M_(x)M_{x} form a set {0}xx[-1,0[:}\{0\} \times\left[-1,0\left[\right.\right., which has no greatest element, which implies that M_((0,1))M_{(0,1)} has no infimum in LL. Therefore, the inclusion map form bar(C)\bar{C} to LL has no right adjoint. M_(x)M_{x} 的下界形成一个集合 {0}xx[-1,0[:}\{0\} \times\left[-1,0\left[\right.\right. ，该集合没有最大元素，这意味着 M_((0,1))M_{(0,1)} 在 LL 中没有下确界。因此，从 bar(C)\bar{C} 到 LL 的包含映射没有右伴随。
By the aid of the interior operator, or equivalently, the right adjoint of the inclusion map, one can extend the binary operator bar(T)\bar{T} on bar(C)\bar{C} to the whole meet semi-lattice LL. 通过内部算子的帮助，或者等价地，通过包含映射的右伴随，可以将二元算子 bar(T)\bar{T} 在 bar(C)\bar{C} 上扩展到整个交会半格 LL 。
Theorem 3.9. Let LL be a bounded meet semi-lattice such that the inclusion map i: bar(C)↪Li: \bar{C} \hookrightarrow L has a right adjoint g:L longrightarrow bar(C)g: L \longrightarrow \bar{C}, then the tt-norm bar(T)\bar{T} on bar(C)\bar{C} defined by Eq. (2) can be extended to a tt-norm widehat(T)\widehat{T} on L, given by 定理 3.9. 设 LL 是一个有界的交半格，且包含映射 i: bar(C)↪Li: \bar{C} \hookrightarrow L 有一个右伴随 g:L longrightarrow bar(C)g: L \longrightarrow \bar{C} ，则在 bar(C)\bar{C} 上由公式 (2) 定义的 tt -范数 bar(T)\bar{T} 可以扩展为 L 上的 tt -范数 widehat(T)\widehat{T} ，其表达式为
widehat(T)(x,y)={[ bar(T)(g(x)","g(y)),x!=1" and "y!=1],[x^^y,x=1" or "y=1]:}\widehat{T}(x, y)= \begin{cases}\bar{T}(g(x), g(y)) & x \neq 1 \text { and } y \neq 1 \\ x \wedge y & x=1 \text { or } y=1\end{cases}
4. Examples 4. 示例
In this section, we collect some examples of t-norms on bounded meet semi-lattices in the literature which are constructed by the technique of ordinal sums. 在本节中，我们收集了一些文献中关于有界交半格的 t-范数的例子，这些例子是通过序数和的技术构造的。
Example 4.1. Let [0,1][0,1] be the unit interval and UU be an open set contained in [0,1][0,1], then {:U=uuu_(n=1)^(oo)]a_(n),b_(n)[\left.U=\bigcup_{n=1}^{\infty}\right] a_{n}, b_{n}[ with ]a_(n),b_(n)[nn]a_(m),b_(m)[=O/:}] a_{n}, b_{n}[\cap] a_{m}, b_{m}\left[=\emptyset\right. whenever n!=mn \neq m. All the endpoints a_(n)a_{n} and b_(n)b_{n} form a chain CC in [0,1][0,1]. Notice that, generally, a_(n)a_{n} need not has a processor and b_(n)b_{n} need not has a successor in the chain CC. For example, when UU is the complement of the Cantor set in [0,1][0,1], it holds that for each nn, neither a_(n)a_{n} has a processor nor b_(n)b_{n} has a successor. If there is a tt-norm T_(n)T_{n} on each interval [a_(n),b_(n)]\left[a_{n}, b_{n}\right], then bar(C)=[0,1]\bar{C}=[0,1] and the ordinal sum bar(T)\bar{T} defined in Theorem 3.1 coincides with the usual one in [13]. 例 4.1. 设 [0,1][0,1] 为单位区间， UU 为包含在 [0,1][0,1] 中的开集，则 {:U=uuu_(n=1)^(oo)]a_(n),b_(n)[\left.U=\bigcup_{n=1}^{\infty}\right] a_{n}, b_{n}[ 在 ]a_(n),b_(n)[nn]a_(m),b_(m)[=O/:}] a_{n}, b_{n}[\cap] a_{m}, b_{m}\left[=\emptyset\right. 的情况下成立，前提是 n!=mn \neq m 。所有的端点 a_(n)a_{n} 和 b_(n)b_{n} 在 [0,1][0,1] 中形成一个链 CC 。注意，通常情况下， a_(n)a_{n} 不一定有处理器， b_(n)b_{n} 在链 CC 中也不一定有后继。例如，当 UU 是 [0,1][0,1] 中的康托集的补集时，对于每个 nn ，既 a_(n)a_{n} 没有处理器，也 b_(n)b_{n} 没有后继。如果每个区间 [a_(n),b_(n)]\left[a_{n}, b_{n}\right] 上有一个 tt -范数 T_(n)T_{n} ，那么 bar(C)=[0,1]\bar{C}=[0,1] 和定理 3.1 中定义的序数和 bar(T)\bar{T} 与 [13] 中的通常定义一致。
Dvořák and Holčapek [8] introduces the ordinal sum of t-norms on bounded meet semi-lattices which are join of subintervals with all endpoints of subintervals being a finite chain. Dvořák 和 Holčapek [8] 介绍了有界交半格上 t-范数的序和，这些半格是子区间的并，所有子区间的端点构成一个有限链。
Example 4.2. Let LL be a bounded meet semi-lattice such that there is a chain 0=b_(0) < b_(1) < dots < b_(n)=10=b_{0}<b_{1}<\ldots<b_{n}=1 with L=uuu_(i=0)^(n-1)[b_(i),b_(i+1)]L=\bigcup_{i=0}^{n-1}\left[b_{i}, b_{i+1}\right]. If T_(i)T_{i} is a t-norm on [b_(i),b_(i+1)]\left[b_{i}, b_{i+1}\right] for i=0,dots,n-1i=0, \ldots, n-1, then the ordinal sum of t -norms T_(0),dots,T_(n-1)T_{0}, \ldots, T_{n-1} on LL are given by 例 4.2. 设 LL 为一个有界的交半格，且存在一个链 0=b_(0) < b_(1) < dots < b_(n)=10=b_{0}<b_{1}<\ldots<b_{n}=1 使得 L=uuu_(i=0)^(n-1)[b_(i),b_(i+1)]L=\bigcup_{i=0}^{n-1}\left[b_{i}, b_{i+1}\right] 。如果 T_(i)T_{i} 是 [b_(i),b_(i+1)]\left[b_{i}, b_{i+1}\right] 上的 t-范数，适用于 i=0,dots,n-1i=0, \ldots, n-1 ，那么 t-范数的序和 T_(0),dots,T_(n-1)T_{0}, \ldots, T_{n-1} 在 LL 上给出。
widehat(T)(x,y)={[T_(i)(x","y)," if "x","y in[b_(i),b_(i+1)[:}],[x^^y," otherwise "]:}\widehat{T}(x, y)= \begin{cases}T_{i}(x, y) & \text { if } x, y \in\left[b_{i}, b_{i+1}[ \right. \\ x \wedge y & \text { otherwise }\end{cases}
In this case, the chain C={b_(0),b_(1),cdots,b_(n)}C=\left\{b_{0}, b_{1}, \cdots, b_{n}\right\} is finite, hence, it is well ordered. Clearly, L= bar(C)L=\bar{C} and widehat(T)= bar(T)\widehat{T}=\bar{T}. 在这种情况下，链 C={b_(0),b_(1),cdots,b_(n)}C=\left\{b_{0}, b_{1}, \cdots, b_{n}\right\} 是有限的，因此它是良序的。显然， L= bar(C)L=\bar{C} 和 widehat(T)= bar(T)\widehat{T}=\bar{T} 。
Ü. Ertuğrul et al. [9] and Dan et al. [5] provided the class of t-norms on bounded meet semi-lattices as a particular case of our construction, as the next examples shows. Ü. Ertuğrul 等人 [9] 和 Dan 等人 [5] 将有界交半格上的 t-范数类作为我们构造的一个特例，正如下一个例子所示。
Example 4.3. Let LL be a bounded meet semi-lattice, and take a in L\\{0,1}a \in L \backslash\{0,1\}. One has two intervals [0,a][0, a] and [a,1][a, 1] equipped with two t-norms T_(1)T_{1} and T_(2)T_{2} respectively. In this case, C={0,a,1}C=\{0, a, 1\} and the set bar(C)=[0,a]uu[a,1]\bar{C}=[0, a] \cup[a, 1]. The inclusion map i: bar(C)↪Li: \bar{C} \hookrightarrow L has a right adjoint given by 例 4.3. 设 LL 为一个有界的交半格，并取 a in L\\{0,1}a \in L \backslash\{0,1\} 。有两个区间 [0,a][0, a] 和 [a,1][a, 1] ，分别配备两个 t-范数 T_(1)T_{1} 和 T_(2)T_{2} 。在这种情况下， C={0,a,1}C=\{0, a, 1\} 和集合 bar(C)=[0,a]uu[a,1]\bar{C}=[0, a] \cup[a, 1] 。包含映射 i: bar(C)↪Li: \bar{C} \hookrightarrow L 具有一个右伴随。
g(x)={[x," if "x in[a","1]],[x^^a," if "x in L\\[a","1]].:}g(x)=\left\{\begin{array}{ll}
x & \text { if } x \in[a, 1] \\
x \wedge a & \text { if } x \in L \backslash[a, 1]
\end{array} .\right.
The ordinal sum widehat(T):L^(2)longrightarrow L\widehat{T}: L^{2} \longrightarrow L of T_(1)T_{1} and T_(2)T_{2} is given explicitly by 序数和 widehat(T):L^(2)longrightarrow L\widehat{T}: L^{2} \longrightarrow L 的 T_(1)T_{1} 和 T_(2)T_{2} 明确表示为
widehat(T)(x,y)={[T_(1)(a^^x","a^^y)",",a≰x","ay;],[T_(2)(x","y),a <= x < 1","a <= y < 1;],[a^^x,axx","a <= y < 1;],[a^^y,a≰y","a <= x < 1;],[x^^y,x=1" or "y=1.]:}\widehat{T}(x, y)= \begin{cases}T_{1}(a \wedge x, a \wedge y), & a \not \leq x, a \not y ; \\ T_{2}(x, y) & a \leq x<1, a \leq y<1 ; \\ a \wedge x & a \not x x, a \leq y<1 ; \\ a \wedge y & a \not \leq y, a \leq x<1 ; \\ x \wedge y & x=1 \text { or } y=1 .\end{cases}
It can easily be shown that this t -norm coincides with the t -norm TT from Theorem 4 of [9]. 可以很容易地证明，这个 t -范数与[9]中定理 4 的 t -范数 TT 一致。
Example 4.4. Let LL be a bounded meet semi-lattice, and a,b in L\\{0,1}a, b \in L \backslash\{0,1\} with a < ba<b. In this case, bar(C)=[0,a]uu[a,b]uu\bar{C}=[0, a] \cup[a, b] \cup[b,1][b, 1], let WW be a t-norm on [a,b][a, b]. The embedded mapping i: bar(C)↪Li: \bar{C} \hookrightarrow L have a right adjoint given by 例 4.4. 设 LL 为一个有界的交半格， a,b in L\\{0,1}a, b \in L \backslash\{0,1\} 具有 a < ba<b 。在这种情况下， bar(C)=[0,a]uu[a,b]uu\bar{C}=[0, a] \cup[a, b] \cup[b,1][b, 1] ，设 WW 为 [a,b][a, b] 上的 t-范数。嵌入映射 i: bar(C)↪Li: \bar{C} \hookrightarrow L 具有一个右伴随，给出如下：
g(x)={[x," if "x in[b","1]],[x^^b," if "x in[a","1]\\[b","1]","],[x^^a," if "x in L\\[a","1]]:}g(x)= \begin{cases}x & \text { if } x \in[b, 1] \\ x \wedge b & \text { if } x \in[a, 1] \backslash[b, 1], \\ x \wedge a & \text { if } x \in L \backslash[a, 1]\end{cases}
Then, the binary operation widehat(T):L^(2)longrightarrow L\widehat{T}: L^{2} \longrightarrow L is a t-norm defined as follows 然后，二元运算 widehat(T):L^(2)longrightarrow L\widehat{T}: L^{2} \longrightarrow L 被定义为以下的 t-范数
widehat(T)(x,y)={[W(g(x)","g(y))," if "g(x)","g(y)in[a","b]" and "1!in{x","y}],[x^^y," if "1in{x","y}],[g(x)^^g(y)," otherwise. "]:}\widehat{T}(x, y)= \begin{cases}W(g(x), g(y)) & \text { if } g(x), g(y) \in[a, b] \text { and } 1 \notin\{x, y\} \\ x \wedge y & \text { if } 1 \in\{x, y\} \\ g(x) \wedge g(y) & \text { otherwise. }\end{cases}
It can easily be shown that this t -norm coincides with the t -norm TT from Theorem 3.3 of [5]. 可以很容易地证明，这个 t -范数与[5]中定理 3.3 的 t -范数 TT 一致。
Example 4.5. Let LL be a bounded meet semi-lattice, {[a_(n),b_(n)]∣n inZ}\left\{\left[a_{n}, b_{n}\right] \mid n \in \mathbb{Z}\right\} be a family of intervals in LL with b_(n) <= a_(n+1)b_{n} \leq a_{n+1}. Given a family L_(n)sube[a_(n),b_(n)],n inZL_{n} \subseteq\left[a_{n}, b_{n}\right], n \in \mathbb{Z} with both a_(n)inL_(n)a_{n} \in L_{n} and b_(n)inL_(n)b_{n} \in L_{n} for all n inZn \in \mathbb{Z}, let 例 4.5. 设 LL 为一个有界的交半格， {[a_(n),b_(n)]∣n inZ}\left\{\left[a_{n}, b_{n}\right] \mid n \in \mathbb{Z}\right\} 为 LL 中的一个区间族，满足 b_(n) <= a_(n+1)b_{n} \leq a_{n+1} 。给定一个族 L_(n)sube[a_(n),b_(n)],n inZL_{n} \subseteq\left[a_{n}, b_{n}\right], n \in \mathbb{Z} ，对于所有 n inZn \in \mathbb{Z} ，同时具有 a_(n)inL_(n)a_{n} \in L_{n} 和 b_(n)inL_(n)b_{n} \in L_{n} ，设
If widetilde(L)\widetilde{L} is the range of some interior operator on LL, then for each n inZ,L_(n)n \in \mathbb{Z}, L_{n} is clearly the range of some interior operator on the interval [a_(n),b_(n)]\left[a_{n}, b_{n}\right], that is, the inclusion map L_(n)↪[a_(n),b_(n)]L_{n} \hookrightarrow\left[a_{n}, b_{n}\right] has a right adjoint. Thus, one can obtain a t-norm widehat(T)_(n)\widehat{T}_{n} on [a_(n),b_(n)]\left[a_{n}, b_{n}\right] by extending a given t -norm T_(n)T_{n} on L_(n)L_{n} through the right adjoint from [a_(n),b_(n)]\left[a_{n}, b_{n}\right] to L_(n)L_{n}. 如果 widetilde(L)\widetilde{L} 是某个内部算子的范围在 LL 上，那么对于每个 n inZ,L_(n)n \in \mathbb{Z}, L_{n} ，显然是区间 [a_(n),b_(n)]\left[a_{n}, b_{n}\right] 上某个内部算子的范围，也就是说，包含映射 L_(n)↪[a_(n),b_(n)]L_{n} \hookrightarrow\left[a_{n}, b_{n}\right] 具有右伴随。因此，可以通过从 [a_(n),b_(n)]\left[a_{n}, b_{n}\right] 到 L_(n)L_{n} 的右伴随扩展给定的 t-范数 T_(n)T_{n} ，在 [a_(n),b_(n)]\left[a_{n}, b_{n}\right] 上获得 t-范数 widehat(T)_(n)\widehat{T}_{n} 。
then it is clear that widetilde(L)sube widehat(L)\widetilde{L} \subseteq \widehat{L}. Recall that the chain CC consisting of all endpoints of the intervals [a_(n),b_(n)]\left[a_{n}, b_{n}\right]. One can see that 那么很明显 widetilde(L)sube widehat(L)\widetilde{L} \subseteq \widehat{L} 。回想一下链 CC 由所有区间的端点 [a_(n),b_(n)]\left[a_{n}, b_{n}\right] 组成。可以看出
and widehat(L)= bar(C)\widehat{L}=\bar{C} if and only if for every n inZn \in \mathbb{Z}, the interval ]b_(n),a_(n+1)[] b_{n}, a_{n+1}[ in LL is empty. Thus, obviously, it holds that 并且 widehat(L)= bar(C)\widehat{L}=\bar{C} 当且仅当对于每个 n inZn \in \mathbb{Z} ，区间 ]b_(n),a_(n+1)[] b_{n}, a_{n+1}[ 在 LL 中是空的。因此，显然，它成立。
and the both inclusion maps have a right adjoint. 并且这两个包含映射都有一个右伴随。
By Theorem 3.1, one obtains a t-norm bar(T)\bar{T} on bar(C)\bar{C}, 根据定理 3.1，可以在 bar(C)\bar{C} 上获得一个 t-范数 bar(T)\bar{T} 。
bar(T)(x,y)={[ widehat(T)_(n)(x","y)," if "x","y in[a_(n),b_(n)]],[x^^y," otherwise "]:}\bar{T}(x, y)= \begin{cases}\widehat{T}_{n}(x, y) & \text { if } x, y \in\left[a_{n}, b_{n}\right] \\ x \wedge y & \text { otherwise }\end{cases}
which is the ordinal sum of the family of t-norms widehat(T)_(n)\widehat{T}_{n} on [a_(n),b_(n)]\left[a_{n}, b_{n}\right]. One can check that the binary operation bar(T)\bar{T} is closed both in the set widetilde(L)\widetilde{L} and widehat(L)\widehat{L}, and coincides with the t-norm widetilde(T)\widetilde{T} on widetilde(L)\widetilde{L} defined in [18] (Theorem 4.1). 这是在 [a_(n),b_(n)]\left[a_{n}, b_{n}\right] 上对 t-范数家族 widehat(T)_(n)\widehat{T}_{n} 的序和。可以检查二元运算 bar(T)\bar{T} 在集合 widetilde(L)\widetilde{L} 和 widehat(L)\widehat{L} 中都是封闭的，并且在文献[18]中定义的 widetilde(L)\widetilde{L} 上的 t-范数 widetilde(T)\widetilde{T} 是相同的（定理 4.1）。
Now we claim that the inclusion map from bar(C)\bar{C} to LL has a right adjoint g:L longrightarrow bar(C)g: L \longrightarrow \bar{C} if and only if the infimum of M_(x)M_{x} exists in LL when a_(m)xa_{m} \not x for all m inZm \in \mathbb{Z}. We divide the proof of the sufficiency in three cases. 现在我们声称，从 bar(C)\bar{C} 到 LL 的包含映射 g:L longrightarrow bar(C)g: L \longrightarrow \bar{C} 有一个右伴随当且仅当在 LL 中存在 M_(x)M_{x} 的下确界，当 a_(m)xa_{m} \not x 对所有 m inZm \in \mathbb{Z} 成立时。我们将充分性的证明分为三种情况。
(i) There is some n inZn \in \mathbb{Z} such that a_(m) <= xa_{m} \leq x (or b_(m) <= xb_{m} \leq x for all m < nm<n and b_(m)xb_{m} \not x (or a_(m+1)xxa_{m+1} \not x x ) for all m >= nm \geq n. Define g(x)=x^^b_(n)g(x)=x \wedge b_{n} (or g(x)=x^^a_(n+1)g(x)=x \wedge a_{n+1} ), one can see that g(x)g(x) is the infimum of M_(x)M_{x} and for all y in bar(C)y \in \bar{C}, y <= g(x)Longleftrightarrow y <= xy \leq g(x) \Longleftrightarrow y \leq x. (i) 存在一些 n inZn \in \mathbb{Z} 使得 a_(m) <= xa_{m} \leq x (或 b_(m) <= xb_{m} \leq x 对所有 m < nm<n 和 b_(m)xb_{m} \not x (或 a_(m+1)xxa_{m+1} \not x x ) 对所有 m >= nm \geq n 。定义 g(x)=x^^b_(n)g(x)=x \wedge b_{n} (或 g(x)=x^^a_(n+1)g(x)=x \wedge a_{n+1} ), 可以看出 g(x)g(x) 是 M_(x)M_{x} 的下确界，并且对所有 y in bar(C)y \in \bar{C} ， y <= g(x)Longleftrightarrow y <= xy \leq g(x) \Longleftrightarrow y \leq x 。
(ii) For all m inZ,b_(m) <= xm \in \mathbb{Z}, b_{m} \leq x. One has that M_(x)={x}M_{x}=\{x\} and x innnn_(m inZ)[b_(m),1]x \in \bigcap_{m \in \mathbb{Z}}\left[b_{m}, 1\right]. Thus, define g(x)=xg(x)=x and it is the infimum of M_(x)M_{x}. Clearly, one has that for all y in bar(C),y <= x Longleftrightarrow y <= g(x)y \in \bar{C}, y \leq x \Longleftrightarrow y \leq g(x). (ii) 对于所有 m inZ,b_(m) <= xm \in \mathbb{Z}, b_{m} \leq x 。有 M_(x)={x}M_{x}=\{x\} 和 x innnn_(m inZ)[b_(m),1]x \in \bigcap_{m \in \mathbb{Z}}\left[b_{m}, 1\right] 。因此，定义 g(x)=xg(x)=x ，它是 M_(x)M_{x} 的下确界。显然，对于所有 y in bar(C),y <= x Longleftrightarrow y <= g(x)y \in \bar{C}, y \leq x \Longleftrightarrow y \leq g(x) ，都有。
(iii) for all m inZ,a_(m)≰xm \in \mathbb{Z}, a_{m} \not \leq x. If inf M_(x)M_{x} exists in LL, then let g(x)=i n fM_(x)g(x)=\inf M_{x} and g(x) <= a_(m)g(x) \leq a_{m} for all m inZm \in \mathbb{Z}. One can see that g(x)g(x) lies in bar(C)\bar{C} and for all y in bar(C),y <= x Longleftrightarrow y <= g(x)y \in \bar{C}, y \leq x \Longleftrightarrow y \leq g(x). (iii) 对于所有 m inZ,a_(m)≰xm \in \mathbb{Z}, a_{m} \not \leq x 。如果 inf M_(x)M_{x} 存在于 LL 中，则让 g(x)=i n fM_(x)g(x)=\inf M_{x} 和 g(x) <= a_(m)g(x) \leq a_{m} 对于所有 m inZm \in \mathbb{Z} 。可以看出 g(x)g(x) 位于 bar(C)\bar{C} 中，并且对于所有 y in bar(C),y <= x Longleftrightarrow y <= g(x)y \in \bar{C}, y \leq x \Longleftrightarrow y \leq g(x) 。
To check the necessity, let gg be the right adjoint of the inclusion map from bar(C)\bar{C} to LL and xx be an element in LL such that each a_(m)xa_{m} \not x. Since g(x) <= xg(x) \leq x, one has that g(x) <= a_(m)g(x) \leq a_{m} for all m inZm \in \mathbb{Z}, otherwise, a_(m) < g(x) <= xa_{m}<g(x) \leq x for some a_(m)a_{m}, which is a contradiction. Thus, g(x)g(x) is a lower bounds of M_(x)M_{x}. Take a lower bound yy of M_(x)M_{x} in LL, then yy is already in bar(C)\bar{C} since y innnn_(m inZ)[0,a_(m)]y \in \bigcap_{m \in \mathbb{Z}}\left[0, a_{m}\right]. One can easily see that y <= g(x)y \leq g(x) and g(x)g(x) is the infimum of M_(x)M_{x} in LL. 为了检查必要性，设 gg 为从 bar(C)\bar{C} 到 LL 的包含映射的右伴随， xx 为 LL 中的一个元素，使得每个 a_(m)xa_{m} \not x 。由于 g(x) <= xg(x) \leq x ，可以得出对于所有 m inZm \in \mathbb{Z} ，都有 g(x) <= a_(m)g(x) \leq a_{m} ，否则，对于某些 a_(m)a_{m} ，有 a_(m) < g(x) <= xa_{m}<g(x) \leq x ，这就产生了矛盾。因此， g(x)g(x) 是 M_(x)M_{x} 的下界。在 LL 中取 M_(x)M_{x} 的一个下界 yy ，那么 yy 已经在 bar(C)\bar{C} 中，因为 y innnn_(m inZ)[0,a_(m)]y \in \bigcap_{m \in \mathbb{Z}}\left[0, a_{m}\right] 。人们可以很容易地看到 y <= g(x)y \leq g(x) 和 g(x)g(x) 是 LL 中 M_(x)M_{x} 的下确界。
Combining with the Theorem 3.15 in [18] and Theorem 3.5, one can see that the following three statements are equivalent: 结合文献[18]中的定理 3.15 和定理 3.5，可以看出以下三个陈述是等价的：
(1) the inclusion map widehat(widetilde(L))↪L\widehat{\widetilde{L}} \hookrightarrow L has a right adjoint; 包含映射 widehat(widetilde(L))↪L\widehat{\widetilde{L}} \hookrightarrow L 具有右伴随
(2) the inclusion map widetilde(L)↪L\widetilde{L} \hookrightarrow L has a right adjoint; (2) 包含映射 widetilde(L)↪L\widetilde{L} \hookrightarrow L 具有右伴随
(3) the inclusion map bar(C)↪L\bar{C} \hookrightarrow L has a right adjoint. (3) 包含映射 bar(C)↪L\bar{C} \hookrightarrow L 具有右伴随。
Therefore, when any one of the above three conditions holds, one can extend the both t-norms widetilde(T)\widetilde{T} and bar(T)\bar{T} to the whole meet semi-lattice LL through the right adjoint of the inclusion maps as in Theorem 3.3. It should be pointed out that the extension of widetilde(T)\widetilde{T} is exactly the one given in [18] (Theorem 4.3), which is less or equal to the extension of bar(C)\bar{C} given by Theorem 3.9. 因此，当上述三个条件中的任何一个成立时，可以通过包含映射的右伴随将两个 t-范数 widetilde(T)\widetilde{T} 和 bar(T)\bar{T} 扩展到整个交半格 LL ，如定理 3.3 所示。需要指出的是， widetilde(T)\widetilde{T} 的扩展正是 [18] 中给出的（定理 4.3），它小于或等于定理 3.9 给出的 bar(C)\bar{C} 的扩展。
5. Conclusion 5. 结论
In a meet semi-lattice LL, given a family of pairwise non-overlapped intervals [a_(i),b_(i)]\left[a_{i}, b_{i}\right] with all endpoints a_(i)a_{i} and b_(i)b_{i} forming a chain and a t-norm T_(i)T_{i} on each interval [a_(i),b_(i)]\left[a_{i}, b_{i}\right], we construct a global t-norm on LL which coincides with each T_(i)T_{i} on [a_(i),b_(i)]\left[a_{i}, b_{i}\right] locally. Our strategy is to collect all elements comparable with all the endpoints a_(i)a_{i} and b_(i)b_{i}. We obtain a subset bar(C)\bar{C} which is large enough to contain all given intervals [a_(i),b_(i)]\left[a_{i}, b_{i}\right]. Moreover, the binary operation TT in Eq. (1) given by Saminger is a t -norm on bar(C)\bar{C}. By the aid of the right adjoint of the inclusion map i: bar(C)longrightarrow Li: \bar{C} \longrightarrow L, we extend the t -norm on bar(C)\bar{C} to the whole meet semi-lattice LL. 在一个交会半格 LL 中，给定一组成对不重叠的区间 [a_(i),b_(i)]\left[a_{i}, b_{i}\right] ，其所有端点 a_(i)a_{i} 和 b_(i)b_{i} 形成一个链，并且在每个区间 [a_(i),b_(i)]\left[a_{i}, b_{i}\right] 上有一个 t-范数 T_(i)T_{i} ，我们构造一个在 LL 上的全局 t-范数，该范数在 [a_(i),b_(i)]\left[a_{i}, b_{i}\right] 上与每个 T_(i)T_{i} 局部一致。我们的策略是收集所有与所有端点 a_(i)a_{i} 和 b_(i)b_{i} 可比的元素。我们获得一个子集 bar(C)\bar{C} ，该子集足够大，可以包含所有给定的区间 [a_(i),b_(i)]\left[a_{i}, b_{i}\right] 。此外，Saminger 在公式(1)中给出的二元运算 TT 是 bar(C)\bar{C} 上的一个 t-范数。在包含映射 i: bar(C)longrightarrow Li: \bar{C} \longrightarrow L 的右伴随的帮助下，我们将 bar(C)\bar{C} 上的 t-范数扩展到整个交会半格 LL 。
Furthermore, by Proposition 5.2 in [19], collect all elements x in Lx \in L, such that for all i in Ii \in I, 此外，根据[19]中的命题 5.2，收集所有元素 x in Lx \in L ，使得对于所有 i in Ii \in I ，
(a) if xx is incomparable to a_(i)a_{i}, then it is incomparable to all u in[a_(i),b_(i)[:}u \in\left[a_{i}, b_{i}[\right., 如果 xx 与 a_(i)a_{i} 不可比，那么它与所有 u in[a_(i),b_(i)[:}u \in\left[a_{i}, b_{i}[\right. 也不可比
(b) if xx is incomparable to b_(i)b_{i}, then it is incomparable to all {:u in]a_(i),b_(i)\left.u \in\right] a_{i}, b_{i} ], (b) 如果 xx 与 b_(i)b_{i} 不可比较，那么它与所有 {:u in]a_(i),b_(i)\left.u \in\right] a_{i}, b_{i} 也不可比较
then we obtain a subset MM of LL, which contains bar(C)\bar{C} clearly. But it remains open to construct t-norms on LL by the aid of the set MM. 然后我们得到一个 LL 的子集 MM ，显然包含 bar(C)\bar{C} 。但是，利用集合 MM 在 LL 上构造 t-范数仍然是一个未解决的问题。
Declaration of competing interest 竞争利益声明
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. 作者声明，他们没有已知的竞争性财务利益或个人关系，这些关系可能会影响本文所报告的工作。
Data availability 数据可用性
No data was used for the research described in the article. 文章中描述的研究没有使用任何数据。
Acknowledgement 致谢
The authors acknowledge the reviewers and professor Hua-peng Zhang for their valuable suggestions, and acknowledge the support of National Natural Science Foundation of China (12171342). 作者感谢评审和张华鹏教授的宝贵建议，并感谢中国国家自然科学基金（12171342）的支持。
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