Generalized quasiorders: constructions and characterizations
D. Jakubíková-Studenovská, R. Pöschel, S. Radeleczki
TL;DR
The paper extends the theory of quasiorders by introducing generalized quasiorders that satisfy the Xi property, linking preservation of an $m$-ary relation by an $n$-ary operation to unary translations via endomorphisms. It develops a suite of algebraic constructions to create new generalized quasiorders, defines special subclasses such as generalized equivalences and generalized partial orders, and proves a unique decomposition of any generalized quasiorder into a weak generalized partial order on a quotient and a generalized equivalence. It further analyzes generalized quasiorders arising from maximal clones, showing that invariant relations often are generalized quasiorders, especially for equivalence relations and lattice orders, and highlights rectangular algebras as prolific sources of generalized quasiorders. The work also outlines conjectures and open problems, including pp- vs quantifier-free pp-definability in non-lattice partial orders and broader questions about the structure of gQuord lattices and clone lattices, setting the stage for future theoretical developments.
Abstract
Quasiorders $\varrho\subseteq A^{2}$ have the property that an operation $f:A^{n}\to A$ preserves $\varrho$ if and only if each (unary) translation obtained from $f$ is an endomorphism of $ρ$. Generalized quasiorders $ρ\subseteq A^{m} $ are generalizations of (binary) quasiorders sharing the same property. We show how new generalized quasiorders can be obtained from given ones using well-known algebraic constructions. Special generalized quasiorders, as generalized equivalences and (weak) generalized partial orders, are introduced, which extend the corresponding notions for binary relations. It turns out that generalized equivalences can be characterized by usual equivalence relations. Extending some known results of binary quasiorders, it is shown that generalized quasiorders can be ``decomposed'' uniquely into a (weak) generalized partial order and a generalized equivalence. Furthermore, generalized quasiorders of maximal clones determined by equivalence or partial order relations are investigated. If $F=Pol\,\varrho$ is a maximal clone and $\varrho$ an equivalence relation or a lattice order, then every(!) relation in $Inv\, F$ is a generalized quasiorder. Moreover, lattice orders are characterized by this property among all partial orders. Finally we prove that each term operation of a rectangular algebra gives rise to a generalized partial order. Some problems requiring further research are also highlighted.