Some algebraic structures on the generalization general products of monoids and semigroups
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Suha Ahmad Wazzan
Arabian Journal of Mathematics
· Ahmet Sinan Cevik · Firat Ates
Some algebraic structures on the generalization general products of monoids and semigroups
Received: 11 November 2019 / Accepted: 10 August 2020 © The Author(s) 2020
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⊕B 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. Mathematics Subject Classification
20E22; 20F05 · 20L05 · 20M05
1 Introduction and preliminaries The notion of Zappa–Szép products generalizes those of direct and semidirect products; the key property is that every element of the Zappa–Szép product can be written uniquely as a product of two elements, one from each factor, in any given order. In the literature, there are some key stone studies on the general product which is also referred as bilateral semidirect products (see [11]), Zappa products (see [7,12,16,18]) or knit products (see [1,14]). As a next step of general product, in [4], the same authors of this paper have recently introduced the generalization of the general product under the name of a higher version of Zappa products for monoids as in the following: For arbitrary monoids A and B, it is known that the A×B denotes the Cartesian product of the number of B copies of the monoid A while the set A⊕B denotes the corresponding direct product. Then a generalization of the general products (both restricted and unrestricted) of the monoid A⊕B by the monoid B ⊕A is defined on ×B ×A ⊕B ⊕A h f A × B and A × B , respectively, with the multiplication ( f, h) f , h = f f , h h , where f, f ∈ A⊕B , h, h ∈ B ⊕A , δ : B ⊕A −→ τ A⊕B , f δh =h f and ψ : A⊕B −→ τ B ⊕A , (h) ψ f = h f S. A. Wazzan (B) · A. S. Cevik Department of Mathematics, KAU King Abdulaziz University, Science Faculty, 21589 Jeddah, Saudi Arabia E-mail: [email protected] A. S. Cevik Department of Mathematics, Selcuk University, Science Faculty, Campus, 42075 Konya, Turkey E-mail: [email protected] F. Ates Department of Mathematics, Balikesir University, Science and Art Faculty, Campus, 10100 Balikesir, Turkey E-mail: [email protected]
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b
are defined by, for a ∈ A and b ∈ B, h f =(h ) f and h f = h f . Also, for x ∈ A and y ∈ B, we define (x) h a = (ax) h and (y)b f = (yb) f such that, for all c ∈ A, d ∈ B, b a (d)(h ) f = dh a f and (c) h f = b f c h a
are held. Moreover, for all f, f ∈ A⊕B and h, h ∈ ⎧ ⎪ p1• : (hh
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