TAILIEUCHUNG - A presentation and some finiteness conditions for a new version of the Schutzenberger product of monoids

In this paper we first define a new version of the Schutzenberger product for any two monoids A and B , and then, by defining a generating and relator set, we present some finite and infinite consequences of the main result. In the final part of this paper, we give necessary and sufficient conditions for this new version to be periodic and locally finite. | Turk J Math (2016) 40: 224 – 233 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article A presentation and some finiteness conditions for a new version of the Sch¨ utzenberger product of monoids 1 ˙ 3,∗, Ismail ˙ ¨ 4 Eylem G¨ uzel KARPUZ1 , Fırat ATES ¸ 2 , Ahmet Sinan C ¸ EVIK Naci CANGUL ¨ Department of Mathematics, Kamil Ozda˘ g Science Faculty, Karamano˘ glu Mehmetbey University, Karaman, Turkey 2 Department of Mathematics, Faculty of Arts and Science, Balıkesir University, Balıkesir, Turkey 3 Department of Mathematics, Faculty of Science, Sel¸cuk University, Konya, Turkey 4 Department of Mathematics, Faculty of Arts and Science, Uluda˘ g University, Bursa, Turkey Received: • Accepted/Published Online: • Final Version: Abstract: In this paper we first define a new version of the Sch¨ utzenberger product for any two monoids A and B , and then, by defining a generating and relator set, we present some finite and infinite consequences of the main result. In the final part of this paper, we give necessary and sufficient conditions for this new version to be periodic and locally finite. Key words: Presentation, Sch¨ utzenberger and wreath products, periodicity, local finiteness 1. Introduction and preliminaries In [4, Theorem , Theorem ], the generator and relator set for the wreath and Sch¨ utzenberger products of arbitrary monoids A and B was defined. Further, in [6, see Theorems and ], the periodicity and local finiteness for semigroups under wreath products were studied. In fact these above results gave us the idea for this paper; since wreath and Sch¨ utzenberger products have been studied a lot for many structures and some important properties have been obtained over them, we wonder what happens if we join both of these products under monoids. Thus, in this paper, we obtain a new monoid (see Section 2) by combining these two .

• Toán học
• Schutzenberger product
• Schutzenberger and wreath products
• Local finiteness
• Finiteness conditions
• Arbitrary monoids