Abstract
The internal Zappa-Szep products emerge when a semigroup has the property that every element has a unique decomposition as a product of elements from two given subsemigroups. The external version constructed from actions of two semigroups on one another satisfying axiom derived by G. Zappa. We illustrate the correspondence between the two versions internal and the external of Zappa-Szep products of semigroups. We consider the structure of the internal Zappa-Szep product as an enlargement. We show how rectangular band can be described as the Zappa-Szep product of a left-zero semigroup and a right-zero semigroup. We find necessary and sufficient conditions for the Zappa-Szep product of regular semigroups to again be regular, and necessary conditions for the Zappa-Szep product of inverse semigroups to again be inverse. We generalize the Billhardt λ-semidirect product to the Zappa-Szep product of a semilattice E and a group G by constructing an inductive groupoid.
Highlights
The Zappa-Szép product of semigroups has two versions internal and external
In [5] we look at Zappa-Szép products derived from group actions on classes of semigroups
A semidirect product of semigroups is an example of a Zappa-Szép product in which one of the actions is taken to be trivial, and semidirect products of semilattices and groups play an important role in the structure theory of inverse semigroups
Summary
The Zappa-Szép product of semigroups has two versions internal and external. In the internal one we suppose that S is a semigroup with two subsemigroups A and B such that each s ∈ S can be written uniquely as s = ab with a ∈ A and b ∈ B. Zappa-Szép products of semigroups provide a rich class of examples of semigroups that include the selfsimilar group actions [2]. [3] uses Li’s construction of semigroup C*-algebras to associate a C*-algebra to Zappa-Szép products and gives an explicit presentation of the algebra They define a quotient C*-algebra that generalises the Cuntz-Pimsner algebras for self-similar actions. They obtain results concerning the extension of (one-sided) congruences, which they apply to (one-sided) congruences on maximal subgroups of regular semigroups They show that a reasonably wide class of D E -simple monoids can be decomposed as Zappa-Szép products. A semidirect product of semigroups is an example of a Zappa-Szép product in which one of the actions is taken to be trivial, and semidirect products of semilattices and groups play an important role in the structure theory of inverse semigroups. Our references for this are [6] and [7]
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