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All Congruence Modular Symmetric and Near-Symmetric Algebras |
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| รหัสดีโอไอ | |
| Creator | 1. Chawewan Ratanaprasert 2. Supharat Thiranantanakorn1 |
| Title | All Congruence Modular Symmetric and Near-Symmetric Algebras |
| Publisher | Silpakorn University Research and Development Institute |
| Publication Year | 2554 |
| Journal Title | Silpakorn University Science and Technology Journal |
| Journal Vol. | 5 |
| Journal No. | 1 |
| Page no. | 24-33 |
| Keyword | Monounary algebra, Congruence distributive, Congruence modular |
| ISSN | 1905-9159 |
| Abstract | For a unary operation f on a finite set A , let denote ?( f ) the least non-negative integer with ( ) ( ) 1 Im Im f f f f ? ? + = which is called the pre-period of f . K. Denecke and S. L. Wismath have characterized all operations f on A with ?( ) 1 f A = ? and prove that ?( ) 1 f A = ? if and only if there exists a d A ? such that 2 A d f d f d = { , ( ), ( ), 1 , ( )} A f d ? ? where 1 ( ) ( ) A A f d f d ? = . C. Ratanaprasert and K. Denecke have characterized all operations f on A with ?( f ) = | A | ?2 for all | A | ? 3; and have characterized all 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 ?( ) {0, 1} f ? for | A | ? 3 which are called symmetric algebras and near-symmetric algebras, respectively. We characterize all operations f whose A is congruence modular. We prove that a symmetric algebra A is congruence 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 a near-symmetric algebra A is congruence modular if and only if ConA is one of the followings: 2 , ? P 2 ( 1), ? ?P 2 , ? L 3 M P? , 3 M P ? ? ( 1), or M L 3 ? where P denote a product of chains and L is a M3 ? head lattice. |
