C. S. Preenu | Searcharxiv

arXiv2024

Semiring arising as Lattice of Groupsemirings

Much study has been done on semigroups which are unions of groups. There are several ways in which a union of groups can be made into a semigroup in which each of the component groups arises as subgroups of the constructed semigroup. An important class of such unions is a semilattice of groups. Group semirings are semirings $(G,+,\cdot )$ where $(G,\cdot )$ is a group and $(G,+)$ is a left zero semigroup. We consider construction of semirings from classes of group semirings $\{G_α:α\in D \}$ indexed by a distributive lattice $D$. It is shown that if $S=\cup\{G_α\}$ is a strong distributive lattice of group semirings $G_α$ then the multiplicative semigroup $(S,\cdot)$ of the semiring $(S,+,\cdot)$ is a Clifford semigroup and the additive semigroup $(S,+)$ is a left normal band. Further in this case all the groups $G_α$ are mutually isomorphic.

math.GR↗

arXiv2022

Cross-connections in Clifford semigroups

An inverse Clifford semigroup (often referred to as just a Clifford semigroup) is a semilattice of groups. It is an inverse semigroup and in fact, one of the earliest studied classes of semigroups. In this short note, we discuss various structural aspects of a Clifford semigroup from a cross-connection perspective. In particular, given a Clifford semigroup, we show that the semigroup of normal cones is isomorphic to the original semigroup, even when it is not a monoid. Hence, we see that cross-connection description degenerates in Clifford semigroups. Further, we specialise the discussion to provide the description of the cross-connection structure in an arbitrary semilattice, also.

math.GR↗