Restriction semigroups and $$\lambda $$ λ -Zappa-Szép products

Abstract

The aim of this paper is to study λ-semidirect and λ-Zappa-Szép products of restriction semigroups. The former concept was introduced for inverse semigroups by Billhardt, and has been extended to some classes of left restriction semigroups. The latter was introduced, again in the inverse case, by Gilbert and Wazzan. We unify these concepts by considering what we name the scaffold of a Zappa-Szép product S T where S and T are restriction. Under certain conditions this scaffold becomes a category. If one action is trivial, or if S is a semilattice and T a monoid, the scaffold may be ordered so that it becomes an inductive category. A standard technique, developed by Lawson and based on the Ehresmann-ScheinNambooripad result for inverse semigroups, allows us to define a product on our category. We thus obtain restriction semigroups that are λ-semidirect products and λ-Zappa-Szép products, extending the work of Billhardt and of Gilbert and Wazzan. Finally, we explicate the internal structure of λ-semidirect products.

Coherency and Constructions for Monoids

“…Also r = 1 because r(aρ) = S × S. We can therefore assume that r = (w, r, ℓ). If w = v, then (w, r, ℓ) = 1(w, r, ℓ) ν (v, 1 M , v)(w, r, ℓ) = 0, proving (10) in this case.…”

“…The next proof, although the calculations are different, follows the exact pattern of the previous, and so we omit it. The monoids in Corollaries 4.10, 4.11 and Remark 4.12 all possess an inverse skeleton [10]; the techniques of [10] would provide an alternative unified approach to these results.…”

“…Finally, we consider the case where 0 appears somewhere in the sequence (9). We claim that (10) (r, 0) ∈ ν.…”

Algebraic properties of Zappa–Szép products of semigroups and monoids

“…This result was mentioned without proof in [10], as an example of a Zappa-Szép product. We include the full argument in this paper for the sake of completeness.…”

Zappa-Szép Products of Semigroups

“…They specifically discuss the Baumslag-Solitar groups, the binary adding machine, the semigroup N N ×  , and the ax b + -semigroup Z Z ×  . In [4] they study semigroups possessing E-regular elements, where an element a of a semigroup S is E-regular if a has an inverse a  such that , aa a a   lie in…”