Minimal but inefficient presentations for semi-direct products of finite cyclic monoids

Abstract

Let G be a semi-direct product of B by A where B and A are both cyclic groups of order n (n ∈ N) and p (any prime), respectively. As a main result of this paper, we prove that G has an inefficient but minimal presentation. Then, as an application of this result, we show that a metacyclic group satisfy the main result. ÖZETB ve A nın her ikiside sırasıyla n (n ∈ N) ve p (asal) mertebeli devirli gruplar olmak üzere, G grubu B nin A ile yarı-direkt çarpımı olsun. Bu çalışmanın ana sonucu olarak G nin etkili olmayan ancak minimal olan bir sunuşa sahip olduğunu ispatladık. Daha sonra bu sonucun bir uygulaması olarak metadevirli grupların bu sonucu sağladığını gösterdik.

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“…In this case, we actually will consider out new product on finite cyclic (monogenic) monoids in which some examples, applications and algebraic structures about these monoids can be found, for instance, in [2]. So let A and B be two such monoids having presentations P A = [x ; x k = x l (k > l)] and P B = [y ; y s = y t (s > t)], respectively.…”

“…Let A and B be monoids. The wreath product A B is regular if and only if A and B are regular, and also for every x ∈ B, y ∈ A, f ∈ A ⊕B and ∈ B ⊕A , there exist e 1 ∈ B and e 2 ∈ A such that e 2 1 = e 1 , e 2 2 = e 2 with (x) f ∈ A(xe 1 ) f and (y) ∈ (e 2 y) B.…”

A presentation and some finiteness conditions for a new version of the Schützenberger product of monoids

“…Here, we will work on finite cyclic monoids, the fundamental facts of which can be found in [3,"Monogenic Semigroups"]. (One can look at the paper [1] for some examples, applications, and algebraic structures on cyclic monoids). Thus let us suppose that A and B are finite cyclic monoids with presentations…”

The Efficiency of the Semi-Direct Products of Free Abelian Monoid with Rank n by the Infinite Cyclic Monoid

“…We should note that since the monoid A is the infinite cyclic monoid, we don't have a trivializer set of D(P A ). Let us consider the presentation P M , as in (1). Then, by [8], a trivializer set of D(P M ) is…”

“…for the monoid M where We may refer [1,2,3,4,5,6,7] to the reader for most of the fundamental material (for instance, semidirect products of monoids, Squier complex, a trivializer set of the Squier complex, spherical and non-spherical monoid pictures) which will be needed here.…”