Adequate Semigroups
Adequate Semigroups
复制标题
DOI：
10.1017/s0013091500016230
复制
发表时间：
1979-06
影响因子：
0.7
通讯作者：
J. Fountain
中科院分区：
文献类型：
--
作者：
J. Fountain
一个幺半群，其中每个主右理想都是投射的，叫做右PP幺半群。在（2），（3），（4）和（8）中研究了这类幺半群的特殊类。右PP幺半群有一个著名的内部刻画，使用关系式 *，定义如下。在半群S上，（a，B）∈ * 当且仅当S的元素a，B通过S的某个超半群中的绿色关系 * 相关联。则幺半群S是右PP幺半群当且仅当S的每个S *-类包含幂等元。单位元的存在性与内部刻画无关，本文研究了幂等元可换且每个幂等元类都含有幂等元的半群类。我们称这样的半群为右适当半群，因为它包含适当的幂等元的充分供给。对偶地，我们可以定义半群上的关系 * 和左适当半群的概念。一个既左适当又右适当的半群称为适当半群。
A monoid in which every principal right ideal is projective is called a right PP monoid. Special classes of such monoids have been investigated in (2), (3), (4) and (8). There is a well-known internal characterisation of right PP monoids using the relation ℒ* which is defined as follows. On a semigroup S, (a,b) ∈ℒ* if and only if the elements a,b of S are related by Green's relation ℒ* in some oversemigroup of S. Then a monoid S is a right PP monoid if and only if each ℒ*-class of S contains an idempotent. The existence of an identity element is not relevant for the internal characterisation and in this paper we study some classes of semigroups whose idempotents commute and in which each ℒ*-class contains an idempotent. We call such a semigroup a right adequate semigroup since it contains a sufficient supply of suitable idempotents. Dually we may define the relation ℛ* on a semigroup and the notion of a left adequate semigroup. A semigroup which is both left and right adequate will be called an adequate semigroup.