Lattices of topologies and quasi-orders on a finite chain

Abstract

The lattice of quasi-orders of the universal algebra 𝐴 is the lattice of those quasi-orders on the set 𝐴 that are compatible with the operations of the algebra, the lattice of the topologies of the algebra is the lattice of those topologies with respect to which the operations of the algebra are continuous. The lattice of quasi-orders and the lattice of topologies of the algebra 𝐴, along with
the lattice of subalgebras and the lattice of congruences, are important characteristics of this algebra. It is known that a lattice of quasi-orders is isomorphically embedded in a lattice that is anti-isomorphic to a lattice of topologies, and in the case of a finite algebra, this embedding is an anti-isomorphism. A chain 𝑋𝑛 of 𝑛 elements is considered as a lattice with operations
𝑥 ∧ 𝑦 = min(𝑥, 𝑦) and 𝑥 ∨ 𝑦 = max(𝑥, 𝑦). It is proved that the lattice of quasi-orders and the lattice of topologies of the chain 𝑋𝑛 are isomorphic to the Boolean lattice of 2^(2𝑛−2) elements. A simple correspondence is found between the quasi-orders of the chain 𝑋𝑛 and words of length 𝑛 − 1 in a 4-letter alphabet. Atoms of the lattice of topologies are found. We deduce from the results on quasi-orders a well-known statement that the congruence lattice of an 𝑛-element chain is Boolean lattioce of 2^(𝑛−1) elements. The results will no longer be true if the chain is considered only with respect to one of the operations ∧,∨.