On representing Mn's by congruence lattice of finite algebras

Abstract

If L is a finite lattice and A is a finite set, then L A = 〈A, L〉 is a finite representation (FR) of L on A iff L is a 0,1-sublattice of E (Λ), the lattice of all equivalence relations on A, and L is isomorphic to L. If L A is a FR of L on A and L is also the congruence lattice of some algebra with base set A, we say L A is a finite algebraic representation (FAR). Thus every finite lattice has a FR, and it is unknown whether every finite lattice has a FAR. Among those finite lattices not known to have FAR's perhaps the M n's (modular lattices of length two with n atoms) provide the most natural class of examples. Some M n's (viz. M p k+1 with p prime) are known to have FAR's. The lattice M 7 is the smallest M n not of that type, and the existence problem remains unsettled for M 7. This paper explores the general structure of FAR's of M n's. We prove: If M n ( n⩾4) has such a representation H A, then M n has another such representation G B satisfying: (i) There are no homomorphisms of G B into G B (as a colored graph) other than automorphisms and constant maps. (ii) G B is vertex transitive. (iii) Each color decomposes B into equivalence classes of uniform size. (iv) Every vertex of H A lies in a subgraph of H A isomorphic to G B. For n⩾4, M n has no such representation h A in which a non-diagonal congruence has a singleton equivalence class. For n ≠ p + 1 ( p prime) M n has no such representation H Λ in which more than one congruence has a two element equivalence class.