On the Pecking Order Between Those of Mitsch and Clifford

Abstract

ABSTRACT Order-theoretic explorations of algebraic structures are known to lead to hitherto hidden insights. Two such relations that have stood out are those of Mitsch and Clifford – the former for the generality in its application and the latter for the insights it offers. In this work, our motivation is to study the converse: we want to explore the extent of the utility of Mitsch’s order and the applicability of Clifford’s order. Firstly, we show that if the Mitsch’s poset is either bounded or a chain, arguably a richer order theoretic structure, the semigroup reduces to one of a simple band. Secondly, noting that the special semigroups on which Clifford’s relation does give rise to an order has not been characterised so far, we solve this problem by proposing a property called Quasi-Projectivity that is essential in this context and also give necessary and sufficient conditions for the Clifford’s relation to give a total and compatible order, even if the semigroup is not commutative. Further, by showing some interesting connections between this relation and the orders obtained by Green’s relations, we further reaffirm the importance and naturalness of the order proposed by Clifford. Finally, by discussing the Clifford’s relations on ordered semigroups, we present some novel perspectives and also show that some of the assumptions in the often cited results of Clifford’s are not necessary. On the whole, our study argues favourably towards Clifford’s than that of the Mitsch’s relation, in so far as the structural information gained about the underlying semigroup.