The 3-forced 2-structures

Abstract

Given sets [Formula: see text] and [Formula: see text], a labeled 2-structure is a function [Formula: see text] from [Formula: see text] to [Formula: see text]. The set [Formula: see text] is called the vertex set of [Formula: see text] and denoted by [Formula: see text]. The label set of [Formula: see text] is the set [Formula: see text] of [Formula: see text] such that [Formula: see text] for some [Formula: see text]. Given [Formula: see text], the 2-substructure [Formula: see text] of [Formula: see text] is denoted by [Formula: see text]. The dual [Formula: see text] of [Formula: see text] is defined on [Formula: see text] as follows. For distinct [Formula: see text], [Formula: see text]. A labeled 2-Structure [Formula: see text] is reversible provided that for [Formula: see text] such that [Formula: see text] and [Formula: see text], if [Formula: see text], then [Formula: see text]. We only consider reversible labeled 2-structures whose vertex set is finite. Let [Formula: see text] and [Formula: see text] be 2-structures such that [Formula: see text]. Given [Formula: see text], [Formula: see text] and [Formula: see text] are [Formula: see text]-hemimorphic if for every [Formula: see text] such that [Formula: see text], [Formula: see text] is isomorphic to [Formula: see text] or [Formula: see text]. Furthermore, let [Formula: see text] be a 2-structure. Given [Formula: see text], [Formula: see text] is [Formula: see text]-forced if [Formula: see text] and [Formula: see text] are the only 2-structures [Formula: see text]-hemimorphic to [Formula: see text]. We characterize the [Formula: see text]-forced 2-structures. Last, we provide a large class of [Formula: see text]-forced 2-structures.