Fregean subtractive varieties with definable congruence (original) (raw)
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On subtractive varieties II: General properties
Algebra Universalis, 1996
As a sequel to we investigate ideal properties focusing on subtractive varieties. After having listed a few basic results, we give several characterizations of the commutator of ideals and prove, for example, that it commutes with finite direct products. We deal with the ideal extension property (IEP) and with related commutator properties, showing for instance that IEP implies that the commutator commutes with restriction to subalgebras. Then we characterize varieties with distributive ideal lattices and relate this property with (a form of) equationally definable principal ideals and with IEP. Then, at the other extreme, we deal with Abelian and Hamiltonian properties (of ideals and congruences), giving for example a purely ideal theoretic characterization of varieties of Abelian groups with linear operations. To finish with, we present a few examples aiming at vindicating our work.
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.
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.
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.
The commutator in equivalential algebras and Fregean varieties
Algebra universalis, 2011
A class K of algebras with a distinguished constant term 0 is called Fregean if congruences of algebras in K are uniquely determined by their 0-cosets and Θ A (0, a) = Θ A (0, b) implies a = b for all a, b ∈ A ∈ K. The structure of Fregean varieties was investigated in a paper by P. Idziak, K. S lomczyńska, and A. Wroński. In particular, it was shown there that every congruence permutable Fregean variety consists of algebras that are expansions of equivalential algebras, i.e., algebras that form an algebraization of the purely equivalential fragment of the intuitionistic propositional logic. In this paper we give a full characterization of the commutator for equivalential algebras and solvable Fregean varieties. In particular, we show that in a solvable algebra from a Fregean variety, the commutator coincides with the commutator of its purely equivalential reduct. Moreover, an intrinsic characterization of the commutator in this setting is given.
Quasi-subtractive varieties: congruences, open filters, and commutators
Quasi-subtractive varieties are a generalisation of subtractive varieties, introduced to account for some correspondence theorems between ideals and congruences in the literature that are not corollaries of general theorems in the theory of subtractive varieties. The main tool for the investigation of quasi-subtractive varieties is the concept of open filter, that extends the notion of ideal due to Gumm and Ursini. In this paper, we explore further the fundamental properties of open filters and congru-ences in this context; moreover, we introduce and investigate a notion of commutator of open filters.
Congruence intersection properties for varieties of algebras
Journal of the Australian Mathematical Society, 1999
It is shown that a variety ν has distributive congruence lattices if and only if the intersection of two principal congruence relations is definable by equations involving terms with parameters. The nature of the terms involved then provides a useful classification of congruence distributive varieties. In particular, the classification puts into proper perspective two stronger properties. A variety is said to have the Principal Intersection Property if the intersection of any two principal congruence relations is principal, or the Compact Intersection Property if the intersection of two compact congruence relations is compact. For non-congruence-distributive varieties, it is shown that some useful constuctions are nevertheless possible.
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 .
The structure of completely meet irreducible congruences in strongly Fregean algebras
Algebra universalis
A strongly Fregean algebra is an algebra such that the class of its homomorphic images is Fregean and the variety generated by this algebra is congruence modular. To understand the structure of these algebras we study the prime intervals projectivity relation in the posets of their completely meet irreducible congruences and show that its cosets have natural structure of Boolean group. In particular, this approach allows us to represent congruences and elements of such algebras as the subsets of upward closed subsets of these posets with some special properties.