Variety of semirings generated by two-element semirings with commutative idempotent multiplication

E. M. Vechtomov ; A. A. Petrov

Čebyševskij sbornik, Tome 15 (2014) no. 3, pp. 12-30

Résumé

The article is devoted to investigation of an variety $\mathfrak N$ generated by two-element commutative multiplicatively idempotent semirings. Two classical theorems of Birkhoff (about the characterization of varieties of algebraic structures, and subdirect reducibility) are initial in the studying of semiring varieties. In 1971 J. A. Kalman proved that there exist up to isomorphism three subdirectly irreducible commutative idempotent semirings satisfying the dual distributive law $x+yz=(x+y)(x+z)$, namely a two-element field, a two-element mono-semiring, and the some three-element semiring. In 1999 S. Ghosh showed that any commutative multiplicatively idempotent semiring with identity $ x + 2xy = x $ is the subdirect product of a Boolean ring and a distributive lattice. In 1992 F. Guzman got a similar result for the variety of all multiplicatively idempotent semirings with zero and unit, satisfying the identity $ 1 + 2x = 1 $. It was proved that every such semiring is commutative. This one is the subdirect product of two-element fields and two-element chains and it may be generated by a single three-element semiring. We obtained the following results in the work. We proved some necessary conditions for subdirect irreducibility of semirings from the variety $\mathfrak M$ of all the semirings with commutative indempotent multiplication. It was shown that an arbitrary semiring from $\mathfrak M$ is subdirect product of two commutative multiplicatively idempotent semirings, one of which has the identity $ 3x = x, $ and the other has the identity $ 3x = 2x. $ We found all the subdirectly irreducible semirings in $\mathfrak N$ and discribed varieties in $\mathfrak N$. It was obtained that in the class $\mathfrak M$ the variety $\mathfrak N$ is defined by the single identity $x+2xy+yz=x+2xz+yz$. We proved that the lattice of all the subvarieties of the variety $\mathfrak N$ is a 16-element Boolean lattice. Bibliography: 16 titles.

Détail

Keywords: semiring, multiplicatively idempotent semiring, the variety of semirings.