M. E. Adams

Open questions related to the problem of Birkhoff and Maltsev

The Birkhoff-Maltsev problem asks for a characterization of those lattices each of which is isomorphic to the lattice L(K) of all subquasivarieties for some quasivariety K of algebraic systems. The current status of this problem, which is still open, is discussed. Various unsolved questions that are related to the Birkhoff-Maltsev problem are also considered, including ones that stem from the theory of propositional logics.

Finite-to-finite universal quasivarieties are Q-universal

Abstract. Let K be a quasivariety of algebraic systems of finite type. K is said to be universal if the category G of all directed graphs is isomorphic to a full subcategory of K. If an embedding of G may be effected by a functor F:G

Congruence relations on de Morgan algebras

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MAXIMAL SUBLATTICES AND FRATTINI SUBLATTICES OF BOUNDED LATTICES

K which assigns a finite algebraic system to each finite graph, then K is said to be finite-to-finite universal. K is said to be Q-universal if, for any quasivariety M of finite type, L(M) is a homomorphic image of a sublattice of L(K), where L(M) and L(K) are the lattices of quasivarieties contained in M and K, respectively.¶We establish a connection between these two, apparently unrelated, notions by showing that if K is finite-to-finite universal, then K is Q-universal. Using this connection a number of quasivarieties are shown to be Q-universal.

The Frattini sublattice of a finite distributive lattice

Various classes of de Morgan algebras whose congruence relations satisfy special conditions are investigated together with their interrelationship. In particular, the classes of congruence permutable, congruence regular, and congruence uniform de Morgan algebras are studied.

QUASIVARIETIES OF IDEMPOTENT SEMIGROUPS

We will address these questions in order, and provide good partial answers, especially for finite lattices which are bounded homomorphic images of a free lattice. Recall that a finite lattice is bounded if and only if it can be obtained from the one element lattice by a sequence of applications of Alan Day’s doubling construction for intervals. In particular, finite distributive lattices are bounded. On the other hand, we do not have a complete solution for any of the above problems. The main results of this paper can be summarized as follows. (1a) For any k > 0, there exists a finite lattice L which has more than |L| maximal sublattices. (1b) A finite bounded lattice L has at most |L| maximal sublattices. (2a) There exist arbitrarily large finite (or even countably infinite) lattices with a maximal sublattice isomorphic to the five element lattice M3. (2b) For any e > 0, there exists a finite bounded lattice L with a maximal sublattice S such that |S| < e|L|. (3a) There exist infinitely many lattice varieties V such that every finite nontrivial lattice L ∈ V is isomorphic to Φ(L′) for some finite lattice L′ ∈ V. (3b) Every finite bounded lattice L can be represented as Φ(K) for some finite bounded lattice K (not necessarily in V(L)).

Endomorphisms and homomorphisms of Heyting algebras

Frattini sublattices of finite distributive lattices are characterized and several applications are given thereof.

A construction for pseudocomplemented semilattices and two applications

It is proved that the lattice L(Bd) of quasivarieties contained in the variety Bd of idempotent semigroups contains an isomorphic copy of the ideal lattice of a free lattice on ω free generators. This result shows that a problem of Petrich [19], which calls for a description of L(Bd), is much more complex than originally expected.

Finite semilattices whose monoids of endomorphisms are regular

For a non-trivial Heyting algebraH, the cardinality of its endomorphism monoid is always at least two. If it is exactly two thenH is called 0-endomorphism rigid. It is shown that there exists a proper class of non-isomorphic 0-endomorphism rigid Heyting algebras. This is a consequence of a more general result: the variety of Heyting algebras is 0-universal.

Homomorphisms of unary algebras with a given quotient

A method is given by which pseudocomplemented semilattices can be constructed from graphs. Two consequences of the method are obtained, namely: there exist continuum-many quasivarieties of pseudocomplemented semilattices; for any non-zero cardinal k , there exist k pairwise non-isomorphic pseudocomplemented semilattices with isomorphic endomorphism monoids.