Equivalence between varieties of square root rings and Boolean algebras with a distinguished automorphism

Abstract

In this paper we study the variety R 2 of square root rings, that is, commutative rings with unit, of characteristic two, with the square root as an additional operation. We prove that this variety is generated by the finite Galois fields GF GF GF(2 k ) and we establish an equivalence between R 2 and the variety BA BA BAδ of Boolean algebras with a distinguished automorphism. Via this equivalence, we will be able to obtain properties of R 2 from the results proved in [M. Abad, J.P. Díaz Varela, M. Zander, Boolean algebras with a distinguished automorphism, Rep. Math. Logic 37 (2003) 101-112].

The semiring variety generated by any finite number of finite fields and distributive lattices

“…Specially, in [10], it is proved that the ring variety generated by a finite ring is finitely based. Thus, in [23] the ring variety of square root rings is considered, and it is proved that it is generated by the finite field F 2 k . In [1] it is proved that the ring variety generated by a finite number of finite fields with pairwise distinct characteristics is finitely based and used in term rewriting.…”

The variety of semirings generated by distributive lattices and finite fields

“…Semisimple varieties occur in the solutions of several natural problems. Thus, in [19] the variety of square root rings are considered, and it is proved that this variety is generated by the finite field F 2 k . In [2] it is proved that the variety generated by finitely many finite fields with pairwise distinct characteristics is finitely based and used in term rewriting.…”