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Construction of Unit Regular Monoids PJAEE, 17 (7) (2020) Construction of Unit Regular Monoids PJAEE, 17 (7) (2020) CONSTRUCTION OF UNIT REGULAR MONOIDS V. K. Sreeja Department of Mathematics, Amrita Vishwa Vidyapeetham , Amritapuri, India V. K. Sreeja: Construction of Unit Regular Monoids - Palarch’s Journal of Archaeology of Egypt / Egyptology, ISSN 1567-214x Key Words : Translational hulls, left translation, right translation, semi group ABSTRACT Let G be the group of units of a monoid S. A unit regular semigroup S should satisfy the condition that corresponding to each 푥 ∈ 푆 there should exist an element 푢 ∈ 퐺 such that 푥 = 푥푢푥 . Consider 푆0 = 푆\퐺 as the set of non units of S. Then evidently 푆0 is a regular sub semigroup of S. Then 푆 = 푆0 ∪ 퐺 푎푛푑 푆0 ∩ 퐺 = ∅. Conversely starting from a regular semigroup So and a group G we have constructed a unit regular semigroup with S with G as group of units of S and 푆0as semigroup of non-units of S. 1. PRELIMINARIES For this construction we have introduced the notion of translational hulls. A right translation of a semigroup S is a transformation ρ satisfying the condition that 푥(푦휌) = (푥푦)ρ for all x, y in S. A left translation of a semigroup S is a transformation λ satisfying the condition that (푥휆)푦 = (푥푦)휆 for all x, y in S. If 푥(푦휆) = (푥휌)푦 for all x, y in S, then we say that a right translation ρ and a left translation λ are linked. Corresponding to each element of the semigroup S we can introduce a transformation 휌푎(휆푎) of S given by 푥휌푎 = 푥푎 [푥휆푎 = 푎푥] for all x in S. If we consider 푇(푆) as the full transformation semigroup on the set S then evidently these transformations are elements of 푇(푆). For any element 푎 ∈ 푆 the inner translations 휌푎 and 휆푎 are linked. Also the set of left (right) translations can be seen to be a sub semigroup of 푇(푆). The translational hull Ω(푆) of a semigroup S is defined to be the set of all ordered pairs(휆, 휌) of linked left and right translations λ and ρ of S We define the translational hull Ω(푆) of a semigroup S to be the set of all pairs (휆, 휌) of linked right and left translations ρ and 휆 of S. If (휆1, 휌1), (휆2, 휌2) ∈ Ω(푆), then so is (휆2휆1, 휌1휌2) . In Ω(푆) we may define a binary operation by, 7855 Construction of Unit Regular Monoids PJAEE, 17 (7) (2020) (휆1, 휌1) (휆2, 휌2) = (휆2휆1, 휌1휌2) Associative property is true and so Ω(푆) is a semigroup. Let Ωo (S) = {(휆푎, 휌푎); 푎 ∈ 푆}. Then Ω0(푆) ⊑ Ω(푆) since 휌푎 and 휆푎 are linked. For any a and b in S, we have (휆푎, 휌푎) (휆푏, 휌푏)= (휆푏휆푎, 휌푎휌푏)= (휆푎푏, 휌푎푏). Starting from a regular semigroup So and a group G, the method of constructing a unit regular semigroup is illustrated in the following theorem. With respect to the translational hull So we give the conditions required for this construction . 2. A CONSTRUCTION Given a regular semigroup and a group, we give the conditions required for the construction of unit regular semigroups . THEOREM 2.1. Let G be a group and 푆0 a regular semigroup. Let the translational hull of the semigroup 푆0be Ω(푆0) and consider Ψ to be a homomorphism from G to Ω(푆0) defined by 휓(푢) = (휓1(푢), 휓2(푢)). Suppose that the following conditions are satisfied by Ψ 휓1(1) and 휓2(1) will act as identity permutations on 푆0. (i) For every 푥 ∈ 푆0, there exists some 푢 ∈ 퐺 such that (푥)휓1(푢) ∈ 퐸(푆0) (or (푥)휓2(푢) ∈ 퐸(푆0) , considering 퐸(푆0) as the set of idempotents of the regular semigroup 푆0 (ii) 휓1(푢1)휓2(푢2) = 휓2(푢2)휓1(푢1) for any two elements 푢1,푢2 ∈ 퐺. Define S to be the disjoint union of 푆0 and G. That is 푆 = 푆0 ∪ 퐺. The binary operation on S can be defined as follows. 푢푥 = (푥)휓1(푢) and 푥푢 = (푥)휓2(푢) for 푥 ∈ 푆0 and 푢 ∈ 퐺. If u and x are elements of G or if they are elements of 푆0, then the product will be same as that in G or 푆0. Then 푆 = 푆0 ∪ 퐺 will be a unit regular semigroup with semigroup of non units as 푆0 and group of units as G. Proof : We will first show that the associative property holds in S. That is we have to prove that (푖)푢(푥푦) = (푢푥)푦 for 푢 ∈ 퐺 and 푥, 푦 ∈ 푆0 (푖푖)(푥푦)푢 = 푥(푦푢) for 푢 ∈ 퐺 and 푥, 푦 ∈ 푆0 (푖푖푖)푥(푢푦) = (푥푢)푦 for 푢 ∈ 퐺 and 푥, 푦 ∈ 푆0 (푖푣)(푢1푢2)푥 = 푢1(푢2푥) for 푢1, 푢2 ∈ 퐺 and 푥 ∈ 푆0 (v)(푥(푢1푢2) = (푥푢1)푢2 for 푢1, 푢2 ∈ 퐺 and 푥 ∈ 푆0 (푣푖)(푢1푥)푢2 = 푢1(푥푢2), for 푢1, 푢2 ∈ 퐺 and 푥 ∈ 푆0 7856 Construction of Unit Regular Monoids PJAEE, 17 (7) (2020) Now since 휓1(푢)is a left translation,(푥휓1(푢))푦 = (푥푦)Ψ1(u) . So 푢(푥푦) = (푢푥)푦 for 푢 ∈ 퐺 and 푥, 푦∈ 푆0. Now since 휓2(푢) is a right translation 푥(푦휓2(푢)) = (푥푦)휓2(푢) .Therefore (푥푦)푢 = 푥(푦푢) for 푢 ∈ 퐺 and 푥, 푦 ∈ 푆0. Because Ψ is a homomorphism we get 휓(푢1푢2) = (휓1(푢1푢2), 휓2(푢1푢2)) . Also we have 휓(푢1)휓(푢2) = (휓1(푢1), 휓2(푢1)) (휓1(푢2), 휓2(푢2)) = (휓1(푢2)휓1(푢1), 휓2(푢1)휓2(푢2)) Hence 휓1(푢1푢2) = 휓1(푢2)휓1(푢1) and 휓2(푢1푢2) = 휓2(푢1)휓2(푢2). So (푥)휓1(푢1푢2) = (푥)[휓1(푢2)휓1(푢1)]. Therefore (푢1푢2)푥 = [(푥)휓1(푢2)]휓1(푢1) = 푢1(푢2푥). In a similar way since 휓2(푢1푢2) = 휓2(푢1)휓2(푢2)we get that 푥(푢1푢2) = (푥푢1)푢2 for 푢1, 푢2 ∈ 퐺 ,푥 ∈ 푆0. Because 휓1(푢) and 휓2(푢) are linked, we get 푥(푦휓1(푢)) = (푥휓2(푢))푦 .So 푥(푢푦) = (푥푢)푦 for 푢 ∈ and 푥, 푦 ∈ 푆0. By condition (iii) we have (푥)[휓1(푢1)휓2(푢2)] =(푥)[휓2(푢2)휓1(푢1)] for 푢1, 푢2 ∈ 퐺 and 푥 ∈ 푆0. Therefore [(푥)휓1(푢1)]휓2(푢2) = [(푥)휓2(푢2)]휓1(푢1) .Hence (푢1푥)푢2 = 푢1(푥푢2). So S is a semigroup. Now since (푥)휓1(1) = 푥 and (푥)휓2(1) = 푥 for any 푥 ∈ 푆0we get that 푥1 = 푥 and 1푥 = 푥 for every x ∈푆0. Therefore the identity element of G will be same as the identity element of S. Hence S is a monoid. Now we show that S is unit regular .For x ∈푆0, let (푥)휓1(푢) ∈ 퐸(푆0) for some element 푢 ∈ 퐺.. Hence , 푢푥 ∈ 퐸(푆0) So (푢푥)(푢푥) = 푢푥. So 푢(푥푢푥) = 푢푥. Therefore 푥푢푥 = 푥. Hence S will be a unit regular semigroup with group of units as G. □ Next we will show that if S is any unit regular semigroup with G as group of units and 푆0as semigroup of non-units then all the requirements of the above theorem are satisfied. THEOREM 2.2. Consider S to be a unit regular semigroup. Then there exists a subgroup G of S and a regular sub semigroup 푆0 of S such that the mapping Ψ from G to Ω(푆0) given by 휓(푢) = (휓1(푢), 휓2 (푢)) is a homomorphism such that (i) 휓1(1)and 휓2(1) act as the identity permutations on 푆0 (ii) (푥)휓1(푢) [ or (x) 휓2 (푢)) ]∈ 퐸(푆0) (ii) 휓1(푢1) 휓2 (푢2) = 휓2 (푢2) 휓1(푢1) for any 푢1, 푢2 ∈ 퐺. Proof : Consider G to be the group of units of S and 푆0 to be the set of all non units of S.If 푢 ∈ 퐺 and 푥 ∈ 푆0, then ux and 푥푢 ∈ 푆0.So we can define two functions such as 휓1(푢) and 휓2(푢) defined by 푥휓1(푢) = 푢푥 and 푥휓2(푢) = 푥푢, for 푥 ∈ 푆0. 7857 Construction of Unit Regular Monoids PJAEE, 17 (7) (2020) Hence 휓1(푢) and 휓2(푢) are respectively the left and right translations of 푆0. Also 휓1(푢) and 휓2(푢) are linked since 푥(푢푦) = (푥푢)푦for 푢 ∈ 퐺 and 푥, 푦 ∈ 푆0 Hence (휓1(푢), 휓2 (푢)) ∈ Ω(푆0) .Now define a function Ψ from G to Ω(푆0) as 휓(푢) = (휓1(푢), 휓2(푢)).Then clearly Ψ is a homomorphism. Since S is unit regular, for any 푥 ∈ 푆 there is some 푢 ∈ 퐺 such that 푥푢푥 = 푥. Hence ux and xu belong to 퐸(푆0).Therefore (푥)휓1(푢) [ or (x) 휓2 (푢)) ]∈ 퐸(푆0) for some 푢 ∈ 퐺. Clearly 휓1(1) and 휓1(1) act as the identity permutations on 푆0. Now for 푢1, 푢2 ∈ 퐺. and 푥 ∈ 푆0 (푢1푥)푢2 = 푢1(푥푢2). Therefore [(푥)휓1(푢1) ] 휓2 (푢2) = [(푥)휓2(푢2) ] 휓1 (푢1) . So, (x) [휓1(푢1) 휓2 (푢2)] = (x) [휓2 (푢2) 휓1(푢1) ] for all 푥 ∈ 푆0. Therefore 휓1(푢1) 휓2 (푢2) = 휓2 (푢2) 휓1(푢1) , REFERENCES • Clifford A.H and Preston G.B., The algebraic theory of semigroups, Surveys of the American Mathematical society 7, Providence, 1961. • K.R. Goodearl, Von Neumann regular rings, (Pitman, 1979). • Hickey J.B and M.V. Lawson, Unit regular monoids, University of Glasgow, Department of Mathematics. • Howie J.M.,An introduction to semigroup Theory, Academic press, New York • Rajan. A.R., On Uniquely unit regular semigroups. Proc. 6th Ramanujan symposium, University of Madras, Chennai (1999), 91-96. • Rajan. A.R., Translational hulls and unit regular semigroups Proc. Of the National seminar on Algebra and Discrete Mathematics.
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