# All Congruence Modular Symmetric and Near-Symmetric Algebras

## Main Article Content

## 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

*M*

_{3}_{ }− 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, M*where

_{3}× P, M_{3}×(P⊕1), or M_{3}× L*P*denote a product of chains and

*L*is a

*M*− head lattice.

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## Article Details

*Science, Engineering and Health Studies*,

*5*(1), 24–33. https://doi.org/10.14456/sustj.2011.2

## References

Denecke, K. and Wismath, S. L. (2002). Universal Algebra and Applications in Theoretical Computer Science. New York: Chapman & Hall.

Jakubikova, D. and Kosice (1982). On congruence relations of unary algebras I. Czechoslovak Mathematical Journal, 32(107): 437-459.

Jakubikova, D. and Kosice (1983). On congruence relations of unary algebras I. Czechoslovak Mathematical Journal, 33(108): 448-466.

McKenzie, R. and Hobby, D. (1998). The structure of finite algebras. Contemporary Mathematics vol. 76, Providence, Rhode Island.

Ratanaprasert, C. and Denecke, K. (2008). Unary operations with long pre-periods. Discrete Mathematics, 308: 4998-5005.