Completions of Generalized Restriction P -Restriction Semigroups
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Completions of Generalized Restriction P-Restriction Semigroups Pan Yan1 · Shoufeng Wang1 Received: 22 May 2019 / Revised: 19 December 2019 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract Generalized restriction P-restriction semigroups are common generalizations of restriction semigroups and generalized inverse ∗-semigroups. Gomes and Szendrei (resp. Ohta and Imaoka) have shown that every restriction semigroup (every generalized inverse ∗-semigroup) can be embedded in a complete, infinitely distributive restriction semigroup (resp. a ∗-complete, infinitely distributive generalized inverse ∗-semigroup). The main aim of this paper is to obtain an entirely corresponding result for generalized restriction P-restriction semigroups. Specifically, among other things, we show that every generalized restriction P-restriction semigroup can be (2,1,1)embedded in a complete, infinitely distributive generalized restriction P-restriction semigroup. Our results generalize and enrich the corresponding results of Gomes, Szendrei, Ohta and Imaoka. Keywords Generalized restriction P-restriction semigroup · Permissible set · Completion Mathematics Subject Classification 20M10
1 Introduction Let S be a semigroup. We denote the set of all idempotents of S by E(S) and the set of all inverses of x ∈ S by V (x). Recall that V (x) = {a ∈ S|xax = x, axa = a} for all x ∈ S. A semigroup S is called regular if V (x) = ∅ for any x ∈ S, and a
Communicated by Peyman Niroomand.
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Shoufeng Wang [email protected] Department of Mathematics, Yunnan Normal University, Kunming 650500, Yunnan, People’s Republic of China
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P. Yan, S. Wang
regular semigroup S is called inverse if E(S) is a commutative subsemigroup (i.e. a subsemilattice) of S, or equivalently, the cardinal of V (x) is equal to 1 for all x ∈ S. Inverse semigroups have been investigated extensively by many authors; see [5,14– 18,21,24,27] for details. In particular, Schein [24] has shown that for a given inverse semigroup S, one can construct a complete, infinitely distributive inverse semigroup C(S) such that S can be embedded into C(S), and C(S) is called a join completion of S (also see Section 1.4 in [16]). Since then, the completions of several classes of inverse semigroups have been studied in the literature; for example, see [2,27]. Moreover, for an inverse semigroup S, by using the semigroup C(S), Lawson [15] has introduced and investigated almost factorizable inverse semigroups (also see Section 7.1 in [16]), and McAlister–Reilly [18] have constructed all E-unitary covers of S through a group G. Moreover, Lawson [14] and Leech [17] have also considered another kind of completions of inverse semigroups, namely meet completions. As a generalization of inverse semigroups in the class of regular semigroups, regular ∗-semigroups have been introduced in Nordahl–Scheiblich [19]. Recall that a semigroup S with a unary operation “◦” is called a regular ∗-semigroup if the following identities hold: x ◦◦ = x, (x y)◦ = y ◦ x ◦ , x x ◦ x
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