Minimal varieties and quasivarieties (original) (raw)
Related papers
Varieties with decidable finite algebras I: Linearity
Algebra Universalis, 1989
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 and .
A Characterization of Finitely Decidable Congruence Modular Varieties
1993
For every nitely generated, congruence modular variety V of - nite type we nd a nite family R of nite rings such that the variety V is nitely decidable if and only if V is congruence permutable and residually small, all solvable congruences in nite algebras from V are Abelian, each congruence above the centralizer of the monolith of a subdirectly irreducible algebra A from V is comparable with all congruences of A, each homomorphic image of a subdirectly irreducible algebra with a non-Abelian monolith has a non-Abelian monolith, and, for each ring R from R ,t he variety of R{modules is nitely decidable.
On the structure of varieties with equationally definable principal congruences III
Algebra Universalis, 1994
p(x,y,z) is a ternary deduction (TD) term function on an algebra A if, for all a, b e A, p(a, b, z) = z (mod O(a, b)), and {p(a, b, z) : z ~ A } is a transversal of the set of equivalence classes of the principal congruence O (a, b). p is commutative if p(a, b, z) and p(a', b', z) define the same transversal whenever O(a, b) = O(a', b 3. P is regular if 6)(p(x, y, 1), 1) = O(x, y) for some constant term 1. The TD term generalizes the (affine) ternary discriminator and is used to investigate the logical structure of nonsemisimple varieties with equationally definable principal congruences (EDPC). Some of the results obtained: The following are equivalent for any variety: (1) "V has a TD term; (2) "//" has EDPC and a certain strong form of the congruence-extension property. If ~e" is semisimple and congruence-permutable, (1) and (2) are equivalent to (3) ~e-is an affine discriminator variety. Afixedpoint ternary discriminator on a set is defined by the conditions: p(x, x, z) = z and, if x # y, p(x, y, z) = d where d is some fixed element; a fixedpoint discriminator variety is defined in analogy to affine discriminator variety. The commutative TD term generalizes the fixedpoint ternary discriminator. The following are equivalent for any semisimple variety: (4) ~g has a commutative TD term; (5) "/f is a fixedpoint discriminator variety. If ~V" is semisimple, congruence-permutable, and has a constant term, (4) and (5) are equivalent to (3); if zv has a second constant term distinct from the first in all nontrivial members of ~, then all five conditions are equivalent to (6) ~e-has a commutative, regular TD term. A hoop is a commutative residuated monoid. Hoops with dual normal operators are defined in analogy with normal Boolean algebras with operators. The main result: A variety of hoops with dual normal operators has a commutative, regular TD term iff it has EDPC iff it has first-order definable principal congruences.
Deductive Varieties of Modules and Universal Algebras
Transactions of the American Mathematical Society, 1985
A variety of universal algebras is called deductive if every subquasivariety is a variety. The following results are obtained: (1) The variety of modules of an Artinian ring is deductive if and only if the ring is the direct sum of matrix rings over local rings, in which the maximal ideal is principal as a left and right ideal. (2) A directly representable variety of finite type is deductive if and only if either (i) it is equationally complete, or (ii) every algebra has an idempotent element, and a ring constructed from the variety is of the form (1) above. This paper initiates a study of varieties of universal algebras with the property that every subquasivariety is a variety. We call such varieties deductive. In §2 it is proved that the variety of left modules of a left Artinian ring R is deductive if and only if R is the direct sum of matrix rings over local rings in which the maximal ideal is principal as a left and right ideal. In §3 we consider varieties of universal algebras of finite type that are directly representable. A variety is directly representable if it is generated by a finite algebra and has, up to isomorphism, only finitely many directly indecomposable finite algebras. Such a variety is deductive if and only if either (i) it is equationally complete, or (ii) every algebra has an idempotent element and a certain ring, constructed from the variety, has the properties listed above.
Reduced sub-powers and the decision problem for finite algebras in arithmetical varieties
Algebra Universalis, 1988
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 monadie second order theory of two successors is applied. 365 366 PAWEL M. IDZIAK ALGEBRA UNIV.
Journal of Symbolic Logic, 2011
Varieties like groups, rings, or Boolean algebras have the property that, in any of their members, the lattice of congruences is isomorphic to a lattice of more manageable objects: e.g. normal subgroups of groups, two-sided ideals of rings, filters (or ideals) of Boolean algebras. Abstract algebraic logic can explain these phenomena at a rather satisfactory level of generality: in every member A of a τ -regular variety V the lattice of congruences of A is isomorphic to the lattice of deductive filters on A of the τ -assertional logic of V. Moreover, if V has a constant 1 in its type and is 1-subtractive, the deductive filters on A ∈ V of the 1-assertional logic of V coincide with the V-ideals of A in the sense of Gumm and Ursini, for which we have a manageable concept of ideal generation.
Congruence quasi-orderability in subtractive varieties
Journal of the Australian Mathematical Society, 2001
In this paper we investigate subtractive varieties of algebras that are congruence quasi-orderable. Though this concept has its origin in abstract algebraic logic, it seems to be worth investigating in a purely algebraic fashion. Besides clarifying the algebraic meaning of this notion, we obtain several structure theorems about such varieties. Also several examples are provided to illustrate the theory.
Axiomatizable and nonaxiomatizable congruence prevarieties
Algebra universalis, 2008
If V is a variety of algebras, let L(V) denote the prevariety of all lattices embeddable in congruence lattices of algebras in V. We give some criteria for the first-order axiomatizability or nonaxiomatizability of L(V). One corollary to our results is a nonconstructive proof that every congruence n-permutable variety satisfies a nontrivial congruence identity.