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