Hereditarily rigid relations

TL;DR

This paper investigates the existence and properties of hereditarily rigid relations on finite sets, establishing their existence, bounds on their arities, and the non-existence of finite hereditarily strongly rigid families, while constructing an infinite one.

Contribution

The paper proves the existence of hereditarily rigid relations, provides a lower bound on their arities, and constructs an infinite hereditarily strongly rigid family, while showing finite such families do not exist.

Findings

Hereditarily rigid relations exist with certain arity bounds.

No finite hereditarily strongly rigid families of relations exist.

An infinite hereditarily strongly rigid family of relations is constructed.

Abstract

An $h$-ary relation $\r$ on a finite set $A$ is said to be \emph{hereditarily rigid} if the unary partial functions on $A$ that preserve $\r$ are the subfunctions of the identity map or of constant maps. A family of relations ${\mathcal F}$ is said to be \emph{hereditarily strongly rigid} if the partial functions on $A$ that preserve every $\r \in {\mathcal F}$ are the subfunctions of projections or constant functions. In this paper we show that hereditarily rigid relations exist and we give a lower bound on their arities. We also prove that no finite hereditarily strongly rigid families of relations exist and we also construct an infinite hereditarily strongly rigid family of relations.