All Congruence Modular Symmetric and Near-Symmetric Algebras

Abstract

For a unary operation f on a finite set A , let denote λ ( f ) the least non-negative integer with Im f ^{λ ( f )} = Im f ^{λ ( f )+1} which is called the pre-period of f . K. Denecke and S. L. Wismath have characterizedall operations f on A with λ ( f ) = A −1 and prove that λ ( f ) = A −1 if and only if there exists a d ∈ Asuch that A = {d, f (d), f ² (d),... , f ^|A|-1(d)} where f ^|A|-1(d) = f ^|A| (d). C. Ratanaprasert and K. Deneckehave characterized all operations f on A with λ ( f ) = | A | −2 for all | A | ≥ 3; and have characterizedall equivalence relations on A which are invariant under f with these long pre-periods.In the paper, we study finite unary algebras A = (A; f ) with λ ( f ) ∈ {0, 1} for | A | ≥ 3 which are called symmetric algebras and near-symmetric algebras, respectively. We characterizeall operations f whose A is congruence modular. We prove that a symmetric algebra A iscongruence modular if and only if the lattice ConA of all congruence relations is either a product of chains or a linear sum of a product of chains with one element top or a M₃ − head lattice; and anear-symmetric algebra A is congruence modular if and only if ConA is one of the followings: 2× P, 2×(P⊕1), 2× L, M₃ × P, M₃ ×(P⊕1), or M₃ × L where P denote a product of chains and L is a M₃ − head lattice.