Structure of lattices of varieties and lattices of quasivarieties: Similarity and difference. I

The aim of the present paper is to provide a unified approach to the study of lattices of varieties and lattices of quasivarieties and to suggest a method for constructing their homomorphic images. We introduce and discuss a (quasi-)Birkhoff class, which is one of the main notions in our paper. In particular, resolved is the problem of Birkhoff concerning characterization of relatively quasiequational classes.Translated fromAlgebra i Logika, Vol. 34, No. 2, pp. 142-168, March-April, 1995.     

Structure of lattices of varieties and lattices of quasivarieties: similarity and difference. III

We give representations for lattices of varieties and lattices of quasivarieties in terms of inverse limits of lattices satisfying a number of additional conditions. Specifically, it is proved that, for any locally finite variety (quasivariety) of algebras V, L _v(V)[resp., L _q(V)] is isomorphic to an inverse limit of a family of finite join semidistributive at 0 (resp., finite lower bounded) lattices. A similar statement is shown to hold for lattices of pseudo-quasivarieties. Various applications are offered; in particular, we solve the problem of Lampe on comparing lattices of varieties with lattices of locally finite ones. Translated fromAlgebra i Logika, Vol. 34, No. 6, pp. 646-666, November-December, 1995.     

Varieties whose skeletons are lattices

Translated from Algebra i Logika, Vol. 31, No. 1, pp. 74–82, January–February, 1992.     

Structure of lattices of varieties and lattices of quasivarieties: Similarity and difference. II

A general characterization of lattices of varieties and lattices of quasivarieties in terms of (quasi)Birkhoff classes is given and a method for constructing their homomorphic images is presented. As an application, it is proved that the lattice of varieties of modular lattices has a complete homomorphism onto the Boolean lattice of subsets of a countable set. Also, sufficient conditions are found for embedding the free lattice with generators in a given lattice of quasivarieties, and we show that these are also sufficient for a quasivariety to be Q-universal. Other applications and examples are given.Translated fromAlgebra i Logika, Vol. 34, No. 4, pp. 369–397, July-August, 1995.     

Algebraic atomistic lattices of quasivarieties

The Gorbunov-Tumanov conjecture on the structure of lattices of quasivarieties is proved true for the case of algebraic lattices. Namely, for an algebraic atomistic lattice L, the following conditions are equivalent: (1) L is represented as L_q(K) for some algebraic quasivariety K; (2) L is represented as S_Λ (A) for some algebraic lattice A which satisfies the minimality condition and nearly satisfies the maximality conditions; (3) L is a coalgebraic lattice admitting an equaclosure operator. Supported by RFFR grants Nos. 96-01-01525 and 96-0-000976, and by DFG grant No. 436 (RUS) 113/2670. Translated from Algebra i Logika, Vol. 36, No. 4, pp. 363–386, July–August, 1997.     

Semigroups whose lattices of right congruences are semiatomic

The main theorems presented here are characterizations of a semigroup with a left identity whose lattice of right congruences is semiatomic. These theorems are preceded by a number of results on minimal right congruences.     

Congruence properties of lattices of quasivarieties

The congruence properties close to being lower boundedness in the sense of McKenzie are treated. In particular, the affirmative answer is obtained to a known question as to whether finite lattices of quasivarieties are lower bounded in the case where quasivarieties are congruence-Noetherian and locally finite. Namely, we state that for every congruence-Noetherian or finitely generated locally finite quasivariety K, the lattice L_q(K) possesses the Day-Pudlak-Tuma property. But if a quasivariety is locally finite without the condition of being finitely generated), that lattice satisfies only the Pudlak-Tuma property. Translated fromAlgebra i Logika, Vol. 36, No. 6, pp. 605–620, Noember, 1997.     

Orthmodular lattices whose MacNeille completions are not orthomodular

The only known example of an orthomodular lattice (abbreviated: OML) whose MacNeille completion is not an OML has been noted independently by several authors, see Adams [1], and is based on a theorem of Ameniya and Araki [2]. This theorem states that for an inner product space V, if we consider the ortholattice ?(V,⊥) = {A \( \subseteq \) V: A = A ^⊥⊥} where A ^⊥ is the set of elements orthogonal to A, then ?(V,⊥) is an OML if and only if V is complete. Taking the orthomodular lattice L of finite or confinite dimensional subspaces of an incomplete inner product space V, the ortholattice ?(V,⊥) is a MacNeille completion of L which is not orthomodular. This does not answer the longstanding question Can every OML be embedded into a complete OML? as L can be embedded into the complete OML ?(V,⊥), where V is the completion of the inner product space V. Although the power of the Ameniya-Araki theorem makes the preceding example elegant to present, the ability to picture the situation is lost. In this paper, I present a simpler method to construct OMLs whose Macneille completions are not orthomodular. No use is made of the Ameniya-Araki theorem. Instead, this method is based on a construction introduced by Kalmbach [7] in which the Boolean algebras generated by the chains of a lattice are glued together to form an OML. A simple method to complete these OMLs is also given. The final section of this paper briefly covers some elementary properties of the Kalmbach construction. I have included this section because I feel that this construction may be quite useful for many purposes and virtually no literature has been written on it.     

Endomorphism monoids of algebras whose subalgebra lattices are chains

A monoidM and a latticeL arealgebraic if there is an algebraA with endomorphism monoid EndA M and subalgebra lattice SuA L. For each chainC we characterize those monoidsM for whichM and C are algebraic. In particular we show that a finite monoidM is algebraic with the three-chain iff the equalizers ofM form a chainE 3. The same assertion however fails for infinite monoids. This generalizes the corresponding result for two-chains and solves a problem posed by B. Jónsson ([2], p. 147). We settle the same question for all longer chainsK. Presented by Ivo Rosenberg.     

On the topological linear spaces whose subspace lattices are modular

LetX be a topological linear space and letL(X) be a lattice of all closed subspaces ofX. We show that in many cases the modularity ofL(X) implies that every bounded subset ofX is finite-dimensional. We derive some topological consequences of the latter result. Due to the significance of the modularity condition forL(X) in quantum axiomatics and elswhere (see [1,14,15]) results also might find application outside the realm of topological linear spaces.