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The flat semirings with nilpotent multiplicative reducts

Zidong Gao, Miaomiao Ren

Abstract

In this paper, we focus on the variety $\mathbf{NF}_3$ generated by all flat semirings with $3$-nilpotent multiplicative reducts. By introducing graph semirings, we characterize all subdirectly irreducible members of $\mathbf{NF}_3$. We prove that the variety $\mathbf{NF}_3$ has uncountably many subvarieties and show that every finitely generated subvariety of $\mathbf{NF}_3$ is a Cross variety. Moreover, we demonstrate that $\mathbf{NF}_3$ has a unique limit subvariety, which is generated by all acyclic graph semirings.

The flat semirings with nilpotent multiplicative reducts

Abstract

In this paper, we focus on the variety generated by all flat semirings with -nilpotent multiplicative reducts. By introducing graph semirings, we characterize all subdirectly irreducible members of . We prove that the variety has uncountably many subvarieties and show that every finitely generated subvariety of is a Cross variety. Moreover, we demonstrate that has a unique limit subvariety, which is generated by all acyclic graph semirings.

Paper Structure

This paper contains 5 sections, 42 theorems, 43 equations, 1 table.

Table of Contents

  1. Introduction and preliminaries
  2. The number of subvarieties of $\mathbf{NF}_3$
  3. The Cross subvarieties of $\mathbf{NF}_3$
  4. The limit subvariety of $\mathbf{NF}_3$
  5. Conclusion

Key Result

Lemma 1.1

The variety ${\bf F}$ is finitely based and each subdirectly irreducible member of ${\bf F}$ is a flat semiring.

Theorems & Definitions (85)

  • Lemma 1.1
  • Proposition 1.3
  • proof
  • Proposition 1.4
  • proof
  • Proposition 1.5
  • proof
  • Example 1.6
  • Lemma 1.7
  • proof
  • ...and 75 more