On nonpermutational transformation semigroups with an application to syntactic complexity

TL;DR

This paper establishes an upper bound on the size of nonpermutational transformation semigroups and applies this result to determine the syntactic complexity of generalized definite languages.

Contribution

It introduces a new upper bound for nonpermutational transformation semigroups and applies it to syntactic complexity in formal language theory.

Findings

Upper bound of n((n-1)!-(n-3)!) for nonpermutational semigroups

Same bound applies to syntactic complexity of generalized definite languages

Provides theoretical limits on transformation semigroup sizes

Abstract

We give an upper bound of $n((n-1)!-(n-3)!)$ for the possible largest size of a subsemigroup of the full transformational semigroup over $n$ elements consisting only of nonpermutational transformations. As an application we gain the same upper bound for the syntactic complexity of (generalized) definite languages as well.