Some algebraic structures on the generalization general products of monoids and semigroups

Abstract

For arbitrary monoids A and B, in Cevik et al. (Hacet J Math Stat 2019:1–11, 2019), it has been recently defined an extended version of the general product under the name of a higher version of Zappa products for monoids (or generalized general product) $$A^{\oplus B}$$ A ⊕ B $$_{\delta }\bowtie _{\psi }B^{\oplus A}$$ δ ⋈ ψ B ⊕ A and has been introduced an implicit presentation as well as some theories in terms of finite and infinite cases for this product. The goals of this paper are to present some algebraic structures such as regularity, inverse property, Green’s relations over this new generalization, and to investigate some other properties and the product obtained by a left restriction semigroup and a semilattice.

The Arithmetic of Generalization for General Products of Monoids

“…The following is general result of [16,Propositions 4.1 and 4.2]. The structure of the principle right, left and two-sided ideals is explained in the following lemma: Lemma 2.…”

“…In particular, we recall the definitions of Green's relations in a monoid S. Define aLb if and only if Sa = Sb, aRb if and only if aS = bS and aJb if and only if SaS = SbS. The relation H = L ∩ R. In [16] we characterize Green's relations for generalized general product.…”

On Classification of Semigroup N by Green’s Theorem

“…Especially, Green's relations may be used to depict the structure of regular semigroups (for example, see eorem 2.1 in [3]). On the other hand, in [4,5], the authors studied some Green's relations for semigroup and monoid structures. e Green's lemma and Green's theorem are the natural next steps in these relations.…”