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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

Abstract 摘要

In this paper, the ordinal sum of a family of given triangular norms on subintervals in a bounded meet semi-lattice is constructed when the subintervals are pairwise non-overlapped and all the endpoints of the subintervals constitute a chain. © 2022 Elsevier B.V. All rights reserved.
在本文中,当子区间成对不重叠且所有子区间的端点构成一个链时,构造了有界交半格上给定三角范数族的序和。© 2022 Elsevier B.V. 保留所有权利。

Keywords: Triangular norm; Ordinal sum; Meet semi-lattice; Galois connection
关键词:三角范数;序和;交半格;伽罗瓦连接

1. Introduction 1. 引言

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 ] [0,1][0,1], ordinal sums play a fundamental role. It is well known that every continuous t-norm on [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1] is an ordinal sum of a family of continuous Archimedean t-norms [ 1 , 13 ] [ 1 , 13 ] [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 ] [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 ] [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 ] [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 t t tt-norms [ 2 , 4 7 , 10 , 12 , 16 21 ] [ 2 , 4 7 , 10 , 12 , 16 21 ] [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 L L LL be a bounded meet semi-lattice and I I II be a totally ordered index set. Further, let ] a i , b i [ i I } a i , b i [ i I } ]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 L L LL and each T i T i T_(i)T_{i} be a t t tt-norm on the interval [ a i , b i ] a i , b i [a_(i),b_(i)]\left[a_{i}, b_{i}\right]. Then the ordinal sum T T 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 t t tt -范数 [ 2 , 4 7 , 10 , 12 , 16 21 ] [ 2 , 4 7 , 10 , 12 , 16 21 ] [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 L L LL 为一个有界交半格, I I II 为一个全序索引集。此外,设 ] a i , b i [ i I } a i , b i [ i I } ]a_(i),b_(i)[∣i in I}:}\left] a_{i}, b_{i}[\mid i \in I\}\right. L L LL 的成对不相交子区间的家族,每个 T i T i T_(i)T_{i} 为区间 [ a i , b i ] a i , b i [a_(i),b_(i)]\left[a_{i}, b_{i}\right] 上的 t t tt -范数。那么序数和 T T TT 由以下公式给出:
T ( x , y ) = { T i ( x , y ) , x , y [ a i , b i ] x y , otherwise T ( x , y ) = T i ( x , y ) ,      x , y a i , b i x y ,       otherwise  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 T T 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 T T 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 t t 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 t t 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 T_(i)T_{i} is to find a t-norm defined globally on the poset L L LL which coincides with each T i T i T_(i)T_{i} on [ a i , b i ] a i , b i [a_(i),b_(i)]\left[a_{i}, b_{i}\right] locally. When the index set I I 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 L L LL does always exist [8]. Moreover, for the index set Z Z Z\mathbb{Z} consisting of all integral numbers, if the endpoints of all intervals [ a i , b i ] a i , b i [a_(i),b_(i)]\left[a_{i}, b_{i}\right] form a chain such that b i a i + 1 b i a i + 1 b_(i) <= a_(i+1)b_{i} \leq a_{i+1} for all i Z i Z i inZi \in \mathbb{Z}, it is shown that the global t-norm on L L LL always exists whenever L L LL is a complete lattice [18]. Generally, the obtained t-norms on L L LL are different from the binary operation given by the Eq. (1).
粗略地说,构造家族 T i T i T_(i)T_{i} 的序和的目的是在偏序集 L L 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 T_(i)T_{i} [ a i , b i ] a i , b i [a_(i),b_(i)]\left[a_{i}, b_{i}\right] 上重合。当索引集 I I 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 L L 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 Z Z\mathbb{Z} ,如果所有区间的端点 [ a i , b i ] a i , b i [a_(i),b_(i)]\left[a_{i}, b_{i}\right] 形成一个链,使得对于所有 i Z i Z i inZi \in \mathbb{Z} 都有 b i a i + 1 b i a i + 1 b_(i) <= a_(i+1)b_{i} \leq a_{i+1} ,则已证明在 L L LL 上的全局 t-范数总是存在,只要 L L LL 是一个完备格[18]。一般来说,在 L L 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 I I II, our strategy is to collect all elements comparable with all the endpoints of the intervals [ a i , b i ] , i I a i , b i , i I [a_(i),b_(i)],i in I\left[a_{i}, b_{i}\right], i \in I. These elements form a subset C ¯ L C ¯ L bar(C)sube L\bar{C} \subseteq L, which contains each subinterval [ a i , b i ] , i I a i , b i , i I [a_(i),b_(i)],i in I\left[a_{i}, b_{i}\right], i \in I. Furthermore, the binary operation T T TT given by Eq. (1) is a t-norm on C ¯ C ¯ bar(C)\bar{C} indeed and coincides with each T i T i T_(i)T_{i} on [ a i , b i ] a i , b i [a_(i),b_(i)]\left[a_{i}, b_{i}\right] locally. Moreover, by aiding of the right adjoint of the inclusion map C ¯ L C ¯ L bar(C)↪L\bar{C} \hookrightarrow L (if it exists), one can extend the t-norm on C ¯ C ¯ bar(C)\bar{C} to the whole bounded meet semi-lattice L L LL.
在本文中,对于任何给定的全序索引集 I I II ,我们的策略是收集与区间 [ a i , b i ] , i I a i , b i , i I [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 C ¯ L C ¯ L bar(C)sube L\bar{C} \subseteq L ,其中包含每个子区间 [ a i , b i ] , i I a i , b i , i I [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 T T TT 确实是 C ¯ C ¯ bar(C)\bar{C} 上的一个 t-范数,并且在 [ a i , b i ] a i , b i [a_(i),b_(i)]\left[a_{i}, b_{i}\right] 上与每个 T i T i T_(i)T_{i} 局部一致。此外,通过辅助包含映射 C ¯ L C ¯ L bar(C)↪L\bar{C} \hookrightarrow L 的右伴随(如果存在),可以将 t-范数从 C ¯ C ¯ bar(C)\bar{C} 扩展到整个有界交半格 L L 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 P P PP is a nonempty set P P PP equipped with an order relation <=\leq (i.e., a reflexive, antisymmetric and transitive binary relation). A poset P P PP is called a totally ordered set, or a chain, if for all a , b P a , b P a,b in Pa, b \in P, it holds that either a b a b a <= ba \leq b or b a b a b <= ab \leq a. The principal ideals of a poset P P PP are the subsets x = { y P y x } x = { y P y x } darr x={y in P∣y <= x}\downarrow x=\{y \in P \mid y \leq x\}. The greatest lower bound of two element x x xx and y y yy is called the meet of x x xx and y y yy, denoted by x y x y x^^yx \wedge y. Dually, the least upper bound of two element of x x xx and y y yy is called the join of x x xx and y y yy, denoted by x y x y x vv yx \vee y. A poset L L LL is called a meet semi-lattice if any two elements x x xx and y y yy have a meet, and is called a lattice if any two elements x x xx and y y yy have both the meet and the join. For a , b L a , b L a,b in La, b \in L, we write a < b a < b a < ba<b for that a b a b a <= ba \leq b but a b a b a!=ba \neq b.
一个偏序集 P P PP 是一个非空集合 P P 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 P a , b P a,b in Pa, b \in P ,要么 a b a b a <= ba \leq b ,要么 b a b a b <= ab \leq a ,则称偏序集 P P PP 为全序集或链。偏序集 P P PP 的主理想是子集 x = { y P y x } x = { y P y x } darr x={y in P∣y <= x}\downarrow x=\{y \in P \mid y \leq x\} 。两个元素 x x xx y y yy 的最大下界称为 x x xx y y yy 的交,记作 x y x y x^^yx \wedge y 。对偶地,两个元素 x x xx y y yy 的最小上界称为 x x xx y y yy 的并,记作 x y x y x vv yx \vee y 。如果任何两个元素 x x xx y y yy 都有交,则称偏序集 L L LL 为交半格;如果任何两个元素 x x xx y y yy 都有交和并,则称为格。对于 a , b L a , b L a,b in La, b \in L ,我们写 a < b a < b a < ba<b 表示 a b a b a <= ba \leq b a b a b a!=ba \neq b
A poset P P 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 L L LL is a bounded meet semi-lattice with the top element 1 and the bottom element 0 .
一个偏序集 P P 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 L L LL 是一个有界的交半格,具有上界元素 1 和下界元素 0 。
Let L L LL be a meet semi-lattice, a , b L a , b L a,b in La, b \in L and a b a b a <= ba \leq b. The subinterval [ a , b ] [ a , b ] [a,b][a, b] is defined as
L L LL 成为一个交会半格, a , b L a , b L a,b in La, b \in L a b a b a <= ba \leq b 。子区间 [ a , b ] [ a , b ] [a,b][a, b] 定义为
[ a , b ] =: { x L a x b } [ a , b ] =: { x L a x b } [a,b]=:{x in L∣a <= x <= b}[a, b]=:\{x \in L \mid a \leq x \leq b\}
and ] a , b [ ] a , b [ ]a,b[] a, b[ is defined as
] a , b [ ] a , b [ ]a,b[] a, b[ 被定义为
] a , b [ =: { x L a < x < b } . ] a , b [ =: { x L a < x < b } . ]a,b[=:{x in L∣a < x < b}.] a, b[=:\{x \in L \mid a<x<b\} .
Two subintervals [ a , b ] [ a , b ] [a,b][a, b] and [ c , d ] [ c , d ] [c,d][c, d] are said to be non-overlapped if ] a , b [ ] c , d [ = ] a , b [ ] c , d [ = ]a,b[nn]c,d[=O/] a, b[\cap] c, d[=\emptyset.
两个子区间 [ a , b ] [ a , b ] [a,b][a, b] [ c , d ] [ c , d ] [c,d][c, d] 被称为不重叠,如果 ] a , b [ ] c , d [ = ] a , b [ ] c , d [ = ]a,b[nn]c,d[=O/] a, b[\cap] c, d[=\emptyset

Obviously, [ a , b ] [ a , b ] [a,b][a, b] is a bounded meet semi-lattice with the top element b b bb and the bottom element a a aa.
显然, [ a , b ] [ a , b ] [a,b][a, b] 是一个有界的交半格,顶元素是 b b bb ,底元素是 a a aa

Definition 2.1 ([6]). Let P P PP be a bounded poset. A binary operation T : P 2 P T : P 2 P T:P^(2)longrightarrow PT: P^{2} \longrightarrow P is called a triangular norm (or a t-norm for short) on P P PP, if for all x , y , z P x , y , z P x,y,z in Px, y, z \in P, it satisfies the following four properties:
定义 2.1 ([6])。设 P P PP 为一个有界偏序集。二元运算 T : P 2 P T : P 2 P 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 P x , y , z P 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 ) 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 ) 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 ) T ( x , z ) T(x,y) <= T(x,z)T(x, y) \leq T(x, z), if y z y z y <= zy \leq z (increasingness);
(2) T ( x , y ) T ( x , z ) T ( x , y ) T ( x , z ) T(x,y) <= T(x,z)T(x, y) \leq T(x, z) ,如果 y z y z 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 ) 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 ) 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 ) = x T ( x , 1 ) = T ( 1 , x ) = x T(x,1)=T(1,x)=xT(x, 1)=T(1, x)=x (neutrality). (4) T ( x , 1 ) = T ( 1 , x ) = x T ( x , 1 ) = T ( 1 , x ) = x T(x,1)=T(1,x)=xT(x, 1)=T(1, x)=x (中立性)。
For a bounded poset P P PP, there is a t -norm, called the drastic product t -norm:
对于一个有界的偏序集 P P PP ,存在一种 t-范数,称为剧烈乘积 t-范数:
T D ( x , y ) = { x y if 1 { x , y } 0 otherwise T D ( x , y ) = x y       if  1 { x , y } 0       otherwise  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 T_(D)T_{D} is the smallest t-norm on P P PP under the pointwise order. If P P PP is a bounded meet semi-lattice, the binary operation ^^\wedge becomes a t-norm, denoted by T M T M T_(M)T_{M}. In this case, T M T M T_(M)T_{M} is the largest t -norm on P P PP under the pointwise order.
剧烈乘积 T D T D T_(D)T_{D} 是在点序下对 P P PP 的最小 t-范数。如果 P P 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 T_(M)T_{M} 。在这种情况下, T M T M T_(M)T_{M} 是在点序下对 P P PP 的最大 t-范数。
In the fuzzy set theory, the unit interval [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1] consisting of real numbers plays the fundamental role among the bounded posets. The t-norm T M T M 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 T_(P)T_{P} and T E T E 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 ] [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 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 T_(P)T_{P} T E T E T_(E)T_{\mathrm{E}}
  • the product t-norm: T P ( x , y ) = x y T P ( x , y ) = x y T_(P)(x,y)=x*yT_{P}(x, y)=x \cdot y.
    产品 t-norm: T P ( x , y ) = x y T P ( x , y ) = x y T_(P)(x,y)=x*yT_{P}(x, y)=x \cdot y
  • the Łukasiewicz t-norm: T Ł ( x , y ) = max { x + y 1 , 0 } T Ł ( x , y ) = max { x + y 1 , 0 } T_(Ł)(x,y)=max{x+y-1,0}T_{\mathrm{Ł}}(x, y)=\max \{x+y-1,0\}Ł.
    Łukasiewicz t-范数: T Ł ( x , y ) = max { x + y 1 , 0 } T Ł ( x , y ) = max { x + y 1 , 0 } T_(Ł)(x,y)=max{x+y-1,0}T_{\mathrm{Ł}}(x, y)=\max \{x+y-1,0\}Ł
Given a family of pairwise disjoint open intervals ] a i , e i [ i I } a i , e i [ i I } ]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 ] [0,1][0,1] and for all i I i I i in Ii \in I, there is a t-norm T i T i T_(i)T_{i} on [ a i , e i a i , e i a_(i),e_(i)a_{i}, e_{i} ], one can constitute a t-norm T T TT as follows [13].
给定单位区间 [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1] 中的一组成对不相交的开区间 ] a i , e i [ i I } a i , e i [ i I } ]a_(i),e_(i)[∣i in I}:}\left] a_{i}, e_{i}[\mid i \in I\}\right. ,对于所有 i I i I i in Ii \in I ,在[ a i , e i a i , e i a_(i),e_(i)a_{i}, e_{i} ]上存在一个 t-范数 T i T i T_(i)T_{i} ,可以如下构造 t-范数 T T TT [13]。
T ( x , y ) = { T i ( x , y ) if ( x , y ) [ a i , e i ] 2 x y otherwise. T ( x , y ) = T i ( x , y )       if  ( x , y ) a i , e i 2 x y       otherwise.  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 T T TT is called the ordinal sums of the family { T i : i I } T i : i I {T_(i):i in I}\left\{T_{i}: i \in I\right\}.
二元运算 T T TT 称为家族 { T i : i I } T i : i I {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 L L LL and a family of pairwise non-overlapped intervals { [ a i , b i ] i I } a i , b i i I {[a_(i),b_(i)]∣i in I}\left\{\left[a_{i}, b_{i}\right] \mid i \in I\right\}, where I I II is a chain such that b i a j b i a j b_(i) <= a_(j)b_{i} \leq a_{j} whenever i j i j i <= ji \leq j, and a t -norm T i T i T_(i)T_{i} on each interval [ a i , b i ] a i , b i [a_(i),b_(i)]\left[a_{i}, b_{i}\right], can one obtain an ordinal sum of the family { T i i I } T i i I {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 L L LL 和一组成对不重叠的区间 { [ a i , b i ] i I } a i , b i i I {[a_(i),b_(i)]∣i in I}\left\{\left[a_{i}, b_{i}\right] \mid i \in I\right\} ,其中 I I II 是一个链,当 b i a j b i a j b_(i) <= a_(j)b_{i} \leq a_{j} i j i j i <= ji \leq j ,以及在每个区间 [ a i , b i ] a i , b i [a_(i),b_(i)]\left[a_{i}, b_{i}\right] 上的 t-范数 T i T i T_(i)T_{i} ,是否可以获得该家族 { T i i I } T i i I {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 L L LL is a bounded meet semi-lattice, I I II is a totally ordered index set and { [ a i , b i ] i I } a i , b i i I {[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 L L LL such that b i a j b i a j b_(i) <= a_(j)b_{i} \leq a_{j} whenever i j i j i <= ji \leq j, that is, the family { [ a i , b i ] i I } a i , b i i I {[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 L L LL 是一个有界的交会半格, I I II 是一个全序的索引集, { [ a i , b i ] i I } a i , b i i I {[a_(i),b_(i)]∣i in I}\left\{\left[a_{i}, b_{i}\right] \mid i \in I\right\} L L LL 中的一组区间,使得 b i a j b i a j b_(i) <= a_(j)b_{i} \leq a_{j} 每当 i j i j i <= ji \leq j 时,即该家族 { [ a i , b i ] i I } a i , b i i I {[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 ] a i , b i [a_(i),b_(i)]\left[a_{i}, b_{i}\right] form a chain in L L LL, which is denoted by