2018/06/28 by Muhammed, P. A. Azeef, Romeo, P. G., Nambooripad, K. S. S.
#18A32 #20M10 #20M50 #Category Theory (math.CT) #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.1806.11031
Cross-connection theory provides the construction of a semigroup from its ideal structure using small categories. A concordant semigroup is an idempotent-connected abundant semigroup whose idempotents generate a regular subsemigroup. We characterize the categories arising from the generalised Green relations in the concordant semigroup as consistent categories and describe their interrelationship using cross-connections. Conversely, given a pair of cross-connected consistent categories, we build a concordant semigroup. We use this correspondence to prove a category equivalence between the category of concordant semigroups and the category of cross-connected consistent categories. In the process, we illustrate how our construction is a generalisation of Nambooripad's cross-connection analysis of regular semigroups. We also identify the inductive cancellative category associated with a pair of cross-connected consistent categories.