Relation categories and coproduct congruence categories in universal algebra

Abstract

It is known that a categoryV-Rel ofadmissible relations can be formed for any variety of algebrasV, such that morphismsA→B correspond to subalgebras ofA x B. We adapt the relation category construction of Hilton and Wu to categoriesC with finite limits and colimits and an image factorization system. The existence ofC-Rel and a dualcograph constructionC-Cogr are proved equivalent to certain stability properties of pullbacks or pushouts forC. For algebraic varietiesV,V -Cogr exists iffV satisfies the amalgamation property (AP) and the congruence extension property (CEP). MorphismsA→B inV-Cogr correspond to congruences on the coproductA + B. It is showed that congruence permutability (CP), the intersection property for amalgamations (IPA), the Hamiltonian property, and the property that congruences 6 are determined by the equivalence class θ[0] can be given characterizations in terms of interlocked pullbacks and pushouts in such a categoryC. A new property IDA (intersections determine amalgamations) is defined, which is dual to CP in this context. Familiar results, such as CP implies congruence modularity, can be proved in such categories. Dually, ifV satisfies AP, CEP, IPA and IDA, it has modular lattices of subalgebras. These results are related to order duality for Su and Con. (For certain varietiesV, the subalgebras ofA are in one-one correspondence with the morphisms below 1A inV -Rel orV-Cogr, and the congruences correspond to the morphisms above 1A.) IfV is pointed (eachA inV has a smallest trivial subalgebra), then a category formulation is obtained for: CP implies the Jonsson-Tarski decomposition properties. The dual shows that pointed varieties satisfying IDA have a restricted form, with pointed unary varieties and varieties ofR-modules as special cases.