The Lattice of Subalgebras of a Boolean Algebra

Abstract

It is well known (1, p. 162) that the lattice of subalgebras of a finite Boolean algebra is dually isomorphic to a finite partition lattice. In this paper we study the lattice of subalgebras of an arbitrary Boolean algebra. One of our main results is that the lattice of subalgebras characterizes the Boolean algebra. In order to prove this result we introduce some notions which enable us to give a characterization and representation of the lattices of subalgebras of a Boolean algebra in terms of a closure operator on the lattice of partitions of the Boolean space associated with the Boolean algebra. Our theory then has some analogy to that of the lattice theory of topological vector spaces. Of some interest is the problem of classification of Boolean algebras in terms of the properties of their lattice of subalgebras, and we obtain some results in this direction.

Maximal subalgebras of Heyting algebras

“…In D. Sachs [13] (see also G. Gratzer, K. M. Koh, and M. Makkai [8]) it is shown inter alia that, for a Boolean algebra with at least eight elements, every non-trivial element is both included and excluded by maximal proper subalgebras. Furthermore, every proper subalgebra is the intersection of maximal subalgebras.…”

On the lattice of subalgebras of a Boolean algebra

“…The second of these results is due to D. Sachs [6] and N. D. Filippov [3]. The first author also gave a characterization of S(B) as a subsystem of an infinite partition lattice.…”

Maximal Subalgebras of MVn-algebras. A Proof of a Conjecture of A. Monteiro

“…When n = 3 we obtain that the maximal subalgebras are those conjectured by A. Monteiro. We also show that every proper subalgebra of an MV n -algebra is an intersection of maximal subalgebras, generalizing a result of Sachs [25] for Boolean algebras (that coincide with MV 2 -algebras).…”

“…Since for each prime filter P of a Boolean algebra ϕ(P ) = P , we obtain the following result of Sachs [25] for Boolean algebras:…”

“…On the other hand, Iturrioz [12], motivated by results of Sachs [25] on maximal subalgebras of Boolean algebras, showed that the subalgebras that we call of type I in this paper, are maximal subalgebras that contain the Boolean elements of three-valued and four-valued Lukasiewicz algebras, and asked whether all maximal subalgebras of these algebras containing the boolean elements are of type I. We also give a positive answer for a class of algebras containing three-valued and four-valued Lukasiewicz algebras (see Theorem 3.2).…”