Endolocality meets homomorphism-homogeneity: a new approach in the study of relational algebras

Abstract

We study the problem of characterizing all relations that can be defined from the fundamental relations of a given relational structure using positive existential formulae. The notion of k-endolocality is introduced in order to measure the complexity of relational structures with respect to this task. The hierarchy of kendolocal structures is thoroughly analysed in algebraic and model-theoretic ways. Interesting cross-connections with homomorphism-homogeneous relational structures are revealed. The interrelations between endolocal relational structures and several model-theoretic notions are collected in the Main Theorem. This Main Theorem is demonstrated to be a useful tool for studying relational algebras and, in particular, weak Krasner algebras. For example, a short proof of F. Börners characterization of weak Krasner clones on a countable set is given.

Towards a Ryll‐Nardzewski‐type theorem for weakly oligomorphic structures

“…The relevance of this notion in the theory of transformation monoids on countable sets was realized quickly [6,7,20,21,22]. Also a classification theory for homomorphism-homogeneous structures emerged quickly [4,8,11,12,16,17,18] and classes of high complexity of finite homomorphism-homogeneous structures were discovered [14,24].…”

“…Moreover, every countable weakly oligomorphic relational structure has a positive existential expansion that is homomorphism‐homogeneous (cf. [, Corollary 4.4], [, Corollary 6.9,Theorem 6.1]). Clearly, every oligomorphic structure is weakly oligomorphic but the reverse does not hold—e.g., there exist countably infinite homomorphism‐homogeneous graphs that have a trivial automorphism group (cf.…”

“…In their seminal paper [5], Peter Cameron and Jaroslav Nešetřil introduced several variations to the concept of homogeneity, one of them being homomorphismhomogeneity-saying that every homomorphism between finitely generated substructures of a given structure extends to an endomorphism of that structure. The relevance of this notion in the theory of transformation monoids on countable sets was realized quickly [6,7,20,21,22]. Also a classification theory for homomorphism-homogeneous structures emerged quickly [4,8,11,12,16,17,18] and classes of high complexity of finite homomorphism-homogeneous structures were discovered [14,24].…”

“…Of particular interest is, to characterize the (countable) structures with an oligomorphic endomorphism monoid. First steps in solving this problem were done in [19] and in [21]. The main goal of this paper is to to understand further the nature of weakly oligomorphic structures, and thus to come closer to a satisfying answer to the problem by Cameron and Nešetřil.…”

On polymorphism-homogeneous relational structures and their clones

“…A relational structure is homomorphism-homogeneous if and only if every homomorphism between finite substructures extends to an endomorphism of the structure. It soon turned out that this notion is very relevant in the theory of transformation monoids on countable sets [8,9,25,26,27]. Moreover, there exists already a rich classification theory for homomorphism-homogeneous structures [4,10,15,16,20,21,22].…”