Finite cyclic semirings with semilattice additive operation defined by two-generated ideal of natural numbers

Abstract

The article deals with finite cyclic semirings with a semilattice addition which are defined as finite cyclic multiplicative monoids~$\langle S, \cdot \rangle$ with an operation of addition~$(+)$ such that the algebraic structure~$\langle S, + \rangle$ is an upper semilattice and laws of distributivity of multiplication over addition are satisfied.

The structure of finite cyclic semirings with a semilattice additive operation defined by a two-generated semiring of nonnegative integers is described.

The result of the work is a theorem about a structure of cyclic semirings with the semilattice additive operation defined by a two-generated ideal of non-negative numbers.
This fact, in particular, allows to calculate the number of cyclic semirings corresponding to each two-generated ideal of non-negative integers.

The method of ideals of a semiring of nonnegative integers is used in the article.
Some properties of ideals of semirings of nonnegative integers determining the structure of finite cyclic semirings are obtained.

This work complements the research of E.\,M. Vechtomova and I.\,V. Orlova where the structure of finite cyclic semirings with~idempotent noncommutative addition is described in terms of cyclic semifields and~finite cyclic semirings with semilattice addition.