On modularity in lattices of congruences on ordered sets
Olav Jordens · Czechoslovak Mathematical Journal · 1992
The following notions are explored in [2].An ordered triple L = (0,R,ar) is called a language, where O and R are pairwise disjoint sets of operation and relation symbols respectively, and ar is the arity function from O U R onto the set of finite cardinals.(It is also specified that for all r G R, we have ar(r) > 0.) An L-model is an ordered triple A = (A',0 A , R A ), where A' is a nonempty set (called the universe of A), 0 A = (o A ; o G O), R A = (r A ; r G R), and for every o G 0,o A is an operation on A' of arity ar(o), and similarly for every r G R, r A is a relation on A' of arity ar(r).The category L (corresponding to L) has all L-models as objects and L-morphisms are maps between the universes of L-models, which preserve both operations and relations in the usual sense.Let X and Y be sets and let /: X -• Y be any map.Then the kernel of map / is defined as ker/ = {{x,y) ; f(x) = f(y)}.For any subcategory K of the category L (corresponding to L) and any A'-object A, the set Con*' A of all A'-congruences on A is defined as the set of all kernels of A'-morphisms from A to any other A'-object B. In this paper we consider the special case where the language L has no operations and only one binary relation, and consider the full subcategory K which consists of all ordered sets (i.e.sets endowed with a reflexive, antisymmetric, and transitive relation).We will always write A = (A', ^.A ) for a poset where A' is the underlying set of A. Thus we have: