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Identity bases for finite cyclic semigroups

Mikhail V. Volkov

Abstract

We provide explicit identity bases for finite cyclic semigroups.

Identity bases for finite cyclic semigroups

Abstract

We provide explicit identity bases for finite cyclic semigroups.

Paper Structure

This paper contains 6 sections, 4 theorems, 18 equations, 1 figure.

Table of Contents

  1. Background
  2. Cyclic semigroups
  3. Words and identities
  4. Identities of finite cyclic semigroups
  5. Identity bases of finite cyclic semigroups
  6. Acknowledgement

Key Result

Lemma 1

A non-balanced identity $u=v$ holds in the nilpotent cyclic semigroup $C_{h,1}$ if and only if either (a)$|u|,|v|\ge h$, or (b)$\operatorname{con}(u)=\operatorname{con}(v)$ and $|u|=|v|\ge h-\min_x\{\operatorname{occ}(x,u),\operatorname{occ}(x,v)\mid \operatorname{occ}(x,u)\ne\operatorname{occ}(x,v)

Figures (1)

  • Figure 1: The Cayley graph of the semigroup $C_{h,d}$

Theorems & Definitions (7)

  • Lemma 1: Lyapin:1979
  • Lemma 2
  • Lemma 3
  • proof
  • Theorem
  • Remark 1
  • proof : Proof of the theorem