Finite Lattice Implication Algebras

Abstract

In this paper, by considering a finite lattice implication algebra L and A ⊆ L, the set of all co-atoms of L, we prove that L is equal to the filter generated by A, that is L = [A). We give a correspondence theorem between the non-trivial minimal filters and co-atoms of L. We prove that if A = {a₁, a₂, ..., a_n}, then L ≌ [a₁) x [a₂) x...x[a_n). Finally, we give a characterization of finite lattice implication algebras. In particular, we show that there exists only one lattice implication algebra of prime order.

Key words and phrases. Finite lattice implication algebras, co-atoms, minimal filter, lattice implication homomorphism.

2000 Mathematics Subject Classification. 06B10, 03G10