On saturated varieties of posemigroups
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Algebra Universalis
On saturated varieties of posemigroups Shabir Ahmad Ahanger, Aftab Hussain Shah and Noor Mohammad Khan Abstract. We show that a permutative variety of posemigroups satisfying a permutation identity x1 x2 · · · xn = xi1 xi2 · · · xin with i1 = 1 and in−1 = n − 1 [in = n and i2 = 2] is saturated if and only if it admits an identity I such that I is not a permutation identity and at least one side of I has no repeated variables. Then we show that the variety of po-rectangular bands is saturated. Finally, we show that a posemigroup S is saturated if the subposemigroup S n , the product of n copies of S, is saturated for some positive integer n. Mathematics Subject Classification. 06F05, 20M07. Keywords. Posemigroup, Dominion, Zigzag Inequalities, Permutative Variety.
1. Introduction and summary In [7] Khan has shown that a permutative variety of semigroups is saturated if and only if it admits an identity I such that I is not a permutation identity and at least one side of I has no repeated variables. In [1] the authors have proved the same result for the variety of commutative posemigroups. In Theorem 3.1, we show that a permutative variety of posemigroups satisfying a permutation identity (1.1) x1 x2 · · · xn = xi1 xi2 · · · xin with i1 = 1 and in−1 = n − 1 [in = n and i2 = 2] is saturated if and only if it admits an identity I such that I is not a permutation identity and at least one side of I has no repeated variables (where the dual statements are Presented by E.W.H. Lee. The second author acknowledges the financial support from Science and Engineering Research Board, Government of India under the Extra Mural Research Grant: EMR/2016/00178. 0123456789().: V,-vol
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enclosed in square brackets). This generalizes the authors result for commutative posemigroups and partially generalizes Khan’s result on permutative variety of semigroups. In [4] Higgins has shown that the variety of normal bands is saturated. This, in particular, shows that the variety of rectangular bands is saturated. In Theorem 3.8, we show that the variety of po-rectangular bands is saturated. Finally, we prove that a posemigroup S is saturated if S n , the subposemigroup of S consisting of all n-fold products of elements of S, is saturated for some positive integer n. This generalizes a result due to Higgins [6] that a semigroup S is saturated if S n is saturated for some positive integer n.
2. Preliminaries A partially ordered semigroup, briefly posemigroup, is a pair (S, ≤) comprising a semigroup S and a partial order ≤ on S that is compatible with its binary operation, i.e. for all s1 , s2 , t1 , t2 ∈ S (s1 ≤ t1 and s2 ≤ t2 ) =⇒ s1 s2 ≤ t1 t2 . If S is a monoid, then we call (S, ≤) a partially ordered monoid, shortly, pomonoid. We call (U, ≤U ) a subposemigroup of posemigroup (S, ≤S ) if U is subsemigroup of the semigroup S and ≤U = ≤S ∩ (U ×U ). The corresponding notion of a subpomonoid is defined analogously. In what follows, we shall denote a posemigroup
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