On finite lattices of topologies of commutative unary algebras
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A. V. Kartashova
Abstract
In this paper, we show that the lattice of topologies of a finite one-generated commutative unary algebra is isomorphic to the lattice of topologies of the characteristic semigroup of this algebra. With the use of this assertion, we give a characterisation of the class of all commutative unary algebras with linearly ordered lattices of topologies. It is proved that if the lattice of congruences or the lattice of topologies of a commutative unary algebra is finite, then the algebra itself is finite. Examples of infinite noncommutative unary algebras with finite lattices of topologies are given. It is proved that for an arbitrary functional signature containing at least one symbol with arity greater than 1 and for any integer n ≥ 2 there exists an infinite algebra of such signature whose lattice of topologies is linearly ordered and consists of n elements.
© de Gruyter 2009
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Articles in the same Issue
- On five types of stability of the lexicographic variant of the combinatorial bottleneck problem
- On the limit distributions of the vertex degrees of conditional Internet graphs
- On the estimation of the periods of noisy binary periodic sequences
- On a constructive approach to the calculation of cardinality of the Ryser classes
- Positively closed classes of three-valued logic generated by one-place functions
- On combinatorial Gray codes with distance 3
- Optimal paths in oriented graphs and eigenvectors in max-⊗ systems
- The factorisation of one-generated partially foliated formations
- On finite lattices of topologies of commutative unary algebras
