Omitting unary and affine types

TL;DR

This paper investigates the conditions under which omitting unary and affine types can be characterized by systems of linear identities involving at most ternary terms, identifying three candidate systems but leaving their effectiveness unresolved.

Contribution

It analyzes all possible linear identity systems on two at most ternary terms, identifying three systems that imply omitting types 1 and 2, and highlights open questions for future research.

Findings

Identified three candidate systems of linear identities

Proved each candidate system implies omitting types 1 and 2

Left open whether these systems fully characterize the omission

Abstract

In this paper we examine the possibility of describing omitting types 1 and 2 by two at most ternary terms and any number of linear identities. All possible cases of systems of linear identities on two at most ternary terms are being analyzed, and it is shown that only three of these systems might describe omitting types 1 and 2. However, we do not resolve whether either of them actually describes omitting mentioned types, but only prove that each of them implies this property, so this question is left for further examination.