Semiring and involution identities of powers of inverse semigroups

The set of all subsets of any inverse semigroup forms an involution semiring under set-theoretical union and element-wise multiplication and inversion. We find structural conditions on a finite inverse semigroup guaranteeing that neither semiring nor involution identities of the involution semiring of its subsets admit a finite identity basis.Comment: 9 page

Representation Theory of Finite Semigroups over Semirings

We develop the representation theory of a finite semigroup over an arbitrary commutative semiring with unit, in particular classifying the irreducible and minimal representations. The results for an arbitrary semiring are as good as the results for a field. Special attention is paid to the boolean semiring, where we also characterize the simple representations and introduce the beginnings of a character theory

Semiring and Involution Identities of Power Groups

For every group G, the set of its subsets forms a semiring under set-theoretical union and element-wise multiplication, and forms an involution semigroup under and element-wise inversion. We show that if the group G is finite, non-Dedekind, and solvable, neither the semiring nor the involution semigroup admits a finite identity basis. We also solve the finite basis problem for the semiring of Hall relations over any finite set. © 2023 The Author(s). Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.Russian Science Foundation, RSF: 22-21-00650Supported by the Russian Science Foundation (grant No. 22-21-00650)