Finite symmetric functions with non-trivial arity gap

TL;DR

This paper investigates symmetric functions with a non-trivial arity gap, analyzing their decomposition and separability properties, which enhances understanding of their structural characteristics in multi-valued logic functions.

Contribution

It provides new results on the decomposition and separability of symmetric functions with non-trivial arity gap, expanding theoretical knowledge in the field.

Findings

Symmetric functions with non-trivial arity gap can be decomposed into minors or subfunctions.

All non-empty sets of essential variables in these functions are separable.

The paper characterizes properties of symmetric functions with arity gap ≥ 2.

Abstract

Given an $n$-ary $k-$valued function $f$, $gap(f)$ denotes the essential arity gap of $f$ which is the minimal number of essential variables in $f$ which become fictive when identifying any two distinct essential variables in $f$. In the present paper we study the properties of the symmetric function with non-trivial arity gap ($2\leq gap(f)$). We prove several results concerning decomposition of the symmetric functions with non-trivial arity gap with its minors or subfunctions. We show that all non-empty sets of essential variables in symmetric functions with non-trivial arity gap are separable.