On Lattices of Topologies of Unary Algebras

Abstract

The ideas of using related algebraic systems for studying properties of the original system go back to E. Galois. Later on, they were developed by R. Dedekind, G. Birkhoff, A. I. Malcev, A. Tarski, A. G. Kurosh, O. Ore, L. A. Skornjakov, G. Gratzer, and other famous algebraists. The last half of the twentieth century was marked by a special attention to lattices associated with algebras. Among them, there are, for example, lattices of subalgebras, congruences, topologies, axiomatizable classes of algebras, etc. These lattices give considerable information about the properties of the original algebras. The traditional problems in this area are the following: the description of algebras whose derived lattices satisfy a given property (e.g., modularity, distributivity), characterizations of lattices represented as a lattice associated with a given class of algebras, and so on. Up to now, substantial progress in solving these problems has been made in classes of lattices associated with unary algebras and especially with unars (a unar is an algebra with one unary operation) (see, e.g., [1–11]). Here the valuable contribution of L. A. Skornjakov to investigating unars should be pointed out ([12]). The lattices of topologies of algebras have been little studied at present. Currently, since the topological algebra is being intensively developed, one can see that this problem is becoming more and more topical ([13]). In the present paper, the problems mentioned above are solved for lattices of topologies of unars. Moreover, we obtain answers to some questions about lattices of topologies of arbitrary algebras naturally arising in this research. The attention to the class of topology lattices of unars is not by chance, since the class of topology lattices on sets coincides with the class of topology lattices of unars 〈A, f〉 of a variety defined by the identity f(x) = x. In this paper, we describe cardinalities of lattices of topologies of uncountable unars. We prove that for the topology of a unar, each of the following conditions implies the finiteness of the unar itself: finiteness of width; descending (ascending) chain condition. But the author [14] proved that for any integer n ≥ 2 and an arbitrary functional signature containing at least one symbol of arity greater than 1, there is a countably infinite algebra of this signature whose topology lattice contains exactly n elements. It is shown that there is no unar whose lattice of topologies is countably infinite. Hence, from the Lowenheim–Skolem–Tarsky theorem, it follows that the class of all lattices of topologies of unars is not axiomatizable. Further, characterizations of those unars whose lattices of topologies are modular, distributive, complemented, pseudocomplemented, Boolean, or chains, respectively, are given. We also describe the class of all modular lattices each of which is isomorphic to a lattice of topologies of some unar. Consequently, the conditions for modularity and distributivity are equivalent for lattices of topologies of unars. We also deal with the upper-subsemilattices 0(A), 1(A), and 2(A) of the lattice (A) of topologies of an arbitrary algebra A whose elements are T0, T1, and T2-topologies on the algebra A, respectively.