Varieties with decidable finite algebras II: Permutability

This paper is a continuation of [3]. Congruence permutability is shown to be a necessary condition for a locally finite congruence distributive variety to have a decidable first order theory of its finite algebras. This is a positive answer to Problem 6 of S. Burns and H. P. Sankappanavar [2]. Moreover this allows us to give a full characterization of finitely generated congruence distributive varieties of finite type with decidable first order theories of their finite members.Presented by Stanley Burris.     

An equational theory for a nilpotent <Emphasis Type="Italic">A</Emphasis>-loop

It is shown that a variety generated by a nilpotent A-loop has a decidable equational (quasiequational ) theory. Thereby the question posed by A. I. Mal’tsev in [6] is answered in the negative, and moreover, a finitely presented nilpotent A-loop has a decidable word problem.     

Varieties with decidable finite algebras I: Linearity

The aim of this paper is to prove that every congruence distributive variety containing a finite subdirectly irreducible algebra whose congruences are not linearly ordered has an undecidable first order theory of its finite members. This fills a gap which kept us from the full characterization of the finitely generated, arithmetical varieties (of finite type) having a decidable first order theory of their finite members. Progress on finding this characterization was made in the papers [14] and [15].Presented by Stanley Burris.     

Amalgamation classes of some distributive varieties

This paper investigates the amalgamation classes of finitely generated varieties with distributive congruence lattices. Necessary and sufficient conditions are given for an algebra to be a member of the amalgamation class of a variety generated by a finite modular lattice or pseudocomplemented distributive lattice and of a filtral variety.Presented by S. Burris.     

Reduced sub-powers and the decision problem for finite algebras in arithmetical varieties

The aim of this paper is to prove that every finitely generated, arithmetical variety of finite type, in which every subdirectly irreducible algebra has linearly ordered congruences has a decidable first order theory of its finite members. The proof is based on a representation of finite algebras from such varieties by some quotients of special subdirect products in which sets of indices are partially ordered into dual trees. Then the result of M. O. Rabin about decidability of the monadic second order theory of two successors is applied.Presented by Stanley Burris.     

Finitely generated varieties of distributive effect algebras

Lattice-ordered effect algebras generalize both MV-algebras and orthomodular lattices. In this paper, finitely generated varieties of distributive lattice effect algebras are axiomatized, and for any positive integer n, the free n-generator algebras in these varieties are described.     

Epimorphisms in Certain Varieties of Algebras

We prove a lemma which, under restrictive conditions, shows that epimorphisms in V are surjective if this is true for epimorphisms from irreducible members of V. This lemma is applied to varieties of orthomodular lattices which are generated by orthomodular lattices of bounded height, and to varieties of orthomodular lattices which are generated by orthomodular lattices which are the horizontal sum of their blocks. The lemma can also be applied to obtain known results for discriminator varieties.     

Amalgamation of Ortholattices

We show that the variety of ortholattices has the strong amalgamation property and that the variety of orthomodular lattices has the strong Boolean amalgamation property, i.e. that two orthomodular lattices can be strongly amalgamated over a common Boolean subalgebra. We give examples to show that the variety orthomodular lattices does not have the amalgamation property and that the variety of modular ortholattices does not even have the Boolean amalgamation property. We further show that no non-Boolean variety of orthomodular lattices which is generated by orthomodular lattices of bounded height can have the Boolean amalgamation property.     

On embedding of finitely generated profinite semigroups into 2-generated profinite semigroups

Koryakov recently asked the following question: when a pseudovariety V does not satisfy any non-trivial identity, does there exist an embedding from any finitely generated V-free profinite semigroup into the 2-generated V-free profinite semigroup? During the conference “Semigroups, Automata and Languages” in Porto (June 1994 [2]), a positive answer to this question was conjectured. We give here a counterexample to this conjecture. Received December 15, 1994; accepted in final form June 5, 1997.     

The free orthomodular lattice on countably many generators is a subalgebra of the free orthomodular lattice on three generators

We show every at most countable orthomodular lattice is a subalgebra of one generated by three elements. As a corollary we obtain that the free orthomodular lattice on countably many generators is a subalgebra of the free orthomodular lattice on three generators. This answers a question raised by Bruns in 1976 [2] and listed as Problem 15 in Kalmbach's book on orthomodular lattices [6]. Received April 12, 2001; accepted in final form May 6, 2002.     

Self-dual bases for varieties of orthomodular lattices

For any finitely based variety of orthomodular lattices, we determine the sizes of all equational bases that are both irredundant and self-dual.     

On a question of Abraham Robinson

In this note we give a negative answer to Abraham Robinson’s question of whether a finitely generated extension of an undecidable field is always undecidable. We construct ‘natural’ undecidable fields of transcendence degree 1 over Q all of whose proper finite extensions are decidable. We also construct undecidable algebraic extensions of Q that allow decidable finite extensions.     

Greechie diagrams of orthomodular partial algebras

Greechie diagrams are well known graphical representations of orthomodular partial algebras, orthomodular posets and orthomodular lattices. For each hypergraph D a partial algebra ⟦D⟧ = (A; ⊕, ′, 0) of type (2,1,0) can be defined. A Greechie diagram can be seen as a special hypergraph: different points of the hypergraph have different interpretations in the corresponding partial algebra ⟦D⟧, and each line in the hypergraph has a maximal Boolean subalgebra as interpretation, in which the points are the atoms. This paper gives some generalisations of the characterisations in [K83] and [D84] of diagrams which represent orthomodular partial algebras (= OMAs), and we give an algorithm how to check whether a given hypergraph D is an OMA-diagram whose maximal Boolean subalgebras are induced by the lines of the hypergraph. Received July 22, 2004; accepted in final form February 1, 2007.     

Orthomodular lattices with rich state spaces

We introduce a new construction technique for orthomodular lattices. In contrast to the preceding constructions, it admits rich spaces of states (= probability measures), i.e., for each pair of incomparable elements a,c there is a state s such that s(a) = 1 > s(c). This allowed a progress in many questions that were open for a long time; among others we prove that there is a continuum of varieties of orthomodular lattices with rich state spaces and solve a problem formulated by R. Mayet in 1985. As a by-product of this research, the uniqueness problem for bounded observables (posed by S. Gudder in 1966) has been solved. As a tool, we introduce also a new construction –identification of atoms in an orthomodular lattice – which may be of separate interest.     

Central elements in pseudoeffect algebras

We introduce the definition of pseudoorthoalgebras and discuss some relationships between orthomodular lattices and pseudoorthoalgebras. Then we study the conditions that a pseudoeffect algebra is isomorphic to an “internal direct product” of ideals generated by orthogonal principal elements. At last, we give some characterizations of central elements in pseudoeffect algebras.     

Varieties ofl-groups are finitely based

We deal with varieties of lattice-ordered groups {ie149-1} defined by the identity [xⁿ, yⁿ]=e. The structure of subdirectly indecomposable l-groups in the variety {ie149-2} is studied, and we establish that l-varieties satisfying the identity [xⁿ, yⁿ]=e and generated by a finitely generated l-group are finitely based. It is shown that l-varieties {ie149-3} with finite axiomatic rank {ie149-4} also have finite bases of identities. Translated fromAlgebra i Logika, Vol. 35, No. 3, pp. 268–287, May–June, 1996.     

A residually finite associative algebra with an undecidable word problem

For any constructive commutative ring k with unity, we furnish an example of a residually finite, finitely generated, recursively defined associative k-algebra with unity whose word problem is undecidable. This answers a question of Bokut’ in [3]. Translated fromAlgebra i Logika, Vol. 39, No. 4, pp. 441–451, July–August, 2000.     

Weakly Orthomodular and Dually Weakly Orthomodular Lattices

We introduce so-called weakly orthomodular and dually weakly orthomodular lattices which are lattices with a unary operation satisfying formally the orthomodular law or its dual although neither boundedness nor complementation is assumed. It turns out that lattices being both weakly orthomodular and dually weakly orthomodular are in fact complemented but the complementation need not be neither antitone nor an involution. Moreover, every modular lattice with complementation is both weakly orthomodular and dually weakly orthomodular. The class of weakly orthomodular lattices and the class of dually weakly orthomodular lattices form varieties which are arithmetical and congruence regular. Connections to left residuated lattices are presented and commuting elements are introduced. Using commuting elements, we define a center of such a (dually) weakly orthomodular lattice and we provide conditions under which such lattices can be represented as a non-trivial direct product.     

Finitary decidability implies congruence permutability for congruence modular varieties

We show that if a locally finite congruence modular varietyV is finitely decidable, thenV has to be congruence permutable.Presented by S. Burris.     

On finitely generated lattices of width four

We define an infinite class ?₄ of infinite lattices with the property that every finitely generated infinite lattice of width four contains (up to duality) a sublattice isomorphic to the herringbone or to a member of ?₄. A consequence is that every finitely generated infinite lattice of width four generates a variety of infinite height (in the lattice of varieties of lattices).