On the Ranks of Semigroups of Transformations on a Finite Set with Restricted Range

• 作者: Jintana Sanwong Vitor H. Fernandes
• 来源期刊: 代数集刊:英文版
• 学科: 数学
• 关键词: 子半群 限制范围 非空子集 格林关系 有限集合 内射 类似
• 年，卷（期）: 2014,（3）
• 页码范围: 497-510
• 页数: 14页
• 分类号: O152.7
• 语种: 中文

摘要

Let ΡΥ(X) be the semigroup of all partial transformations on X, Υ(X) and Ι(X) be the subsemigroups of ΡΥ(X) of all full transformations on X and of all injective partial transformations on X, respectively. Given a non-empty subset Y of X, let ΡΥ(X, Y) = {α∈ ΡΥ(X) : Xα Y}, Υ(X, Y) = ΡΥ(X, Y) ∩Υ(X) and Ι(X, Y) = ΡΥ(X, Y) ∩ Ι(X). In 2008, Sanwong and Sommanee described the largest regular subsemigroup and determined Green's relations of Υ(X,Y). In this paper, we present analogous results for bothΡ Υ(X, Y) and Ι(X, Y). For a finite set X with |x|≥ 3, the ranks of ΡΥ(X) = ΡΥ(X, X), Υ(X) = Υ(X, X) and.Ι(X) = Ι(X, X) are well known to be 4, 3 and 3, respectively. In this paper, we also compute the ranks of ΡΥ(X,Y), Υ(X, Y) and Ι(X, Y) for any proper non-empty subset Y of X.