Definable principal congruences in congruence distributive varieties

We present an algorithm that, given a finite algebraA generating a congruence distributive (CD) variety, determines whether this variety has first order definable principal congruences (DPC). In fact, DPC turns out to be equivalent to the extendability of the principal congruences of certain subalgebras of the algebras in HS(A ³). To verify this algorithm, we investigate combinatorial properties of finite subdirect powers ofA. Our theorem has a relatively simple formulation for arithmetical algebras. As an application, we obtain McKenzie's result that there are no nondistributive lattice varieties with DPC.Presented by A. Pixley.Finally I wish to thank E. Fried, R. W. Quackenbush and P. Pröhle for many helpful conversations (some ideas of the paper came up by considering weakly associative lattices), and to A. F. Pixley for raising the problem mentioned in the first section, which was the starting point of this investigation.     

Definable principal subcongruences

For varieties of algebras, we present the property of having "definable principal subcongruences" (DPSC), generalizing the concept of having definable principal congruences. It is shown that if a locally finite variety V of finite type has DPSC, then V has a finite equational basis if and only if its class of subdirectly irreducible members is finitely axiomatizable. As an application, we prove that if A is a finite algebra of finite type whose variety V(A) is congruence distributive, then V(A) has DPSC. Thus we obtain a new proof of the finite basis theorem for such varieties. In contrast, it is shown that the group variety V(S ₃ ) does not have DPSC. Received May 9 2000; accepted in final form April 26, 2001.     

Operator properties of congruence permutable varieties with strongly definable principal congruences

In an attempt to describe the partially ordered monoid of operators generated by the operators H (homomorphic images), S (subalgebras), \({P_{\rm f}}\) (filtered products) for the variety \({\mathcal{R}_{\rm c}}\) of commutative rings, several results about congruence permutable varieties have been discovered.     

On the structure of varieties with equationally definable principal congruences III

p(x, y, z) is aternary deduction (TD) term function on an algebra A if, for alla, b A, p(a, b,z) z (mod (a, b)), and {p(a, b, z): z A} is a transversal of the set of equivalence classes of the principal congruence (a, b). p iscommutative ifp(a, b, z) and p(a', b', z) define the same transversal whenever0(a, b)=0(a', b'). p isregular if(p(x, y, 1), 1)=0(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)V has EDPC and a certain strong form of the congruence-extension property. IfV is semisimple and congruence-permutable, (1) and (2) are equivalent to (3)V is an affine discriminator variety. Afixedpoint ternary discriminator on a set is defined by the conditions:p(x, x,z)=z and, ifx y, p(x, y,z)=d whered is some fixed element; afixedpoint 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)V has a commutative TD term; (5)V is a fixedpoint discriminator variety. IfV is semisimple, congruence-permutable, and has a constant term, (4) and (5) are equivalent to (3); ifV has a second constant term distinct from the first in all nontrivial members ofV then all five conditions are equivalent to (6)V has a commutative, regular TD term. Ahoop 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.Dedicated to Bjarni Jónsson on his 70th birthdayPresented by R. W. Quackenbush.The authors gratefully acknowledge the support of National Science Foundation Grants DMS-8703743 and DMS-8805870.     

On the structure of varieties with equationally definable principal congruences IV

The notion of apseudo-interior algebra is introduced; it is a hybrid of a (topological) interior algebra and a residuated partially ordered monoid. The elementary arithmetic of pseudo-interior algebras is developed leading to a simple equational axiomatization. A notion ofopen filter analogous to the open filters of interior algebras is investigated. Pseudo-interior algebras represent, in algebraic form, the logic inherent in varieties with acommutative, regular ternary deductive (TD) term p(x, y, z), which is defined by the conditions: (1)p(x,y,z) z (mod(x, y)); (2) for fixed elementsa, b of an algebra A, {p(a, b, z):z A} is a transversal of the set of equivalence classes of (a, b); (3)p(a, b, z) andp(a,b,z) define the same transversal whenever(a,b)=(a,b); (4)(p(x, y, 1), 1)= (x, y) for some constant term 1. The TD term generalizes the (affine) ternary discriminator. Varieties with a commutative, regular TD term include most of the varieties of traditional algebraic logic as well as all double-pointed affine discriminator varieties andn-potent hoops (residuated commutative po-monoids in which the partial ordering is inverse divisibility). The main theorem:A variety has a commutative, regular TD term iff it is termwise definitionally equivalent to a pseudo-interior algebra with additional operations that are compatible with the open filters in a natural way.Presented by R. W. Quackenbush.The authors gratefully acknowledge the support of National Science Foundation Grants DMS-8703743 and DMS-8805870.     

Para primal varieties: A study of finite axiomatizability and definable principal congruences in locally finite varieties

We introduce new sufficient conditions for a finite algebraU to possess a finite basis of identities. The conditions are that the variety generated byU possess essentially only finitely many subdirectly irreducible algebras, and have definable principal congruences. Both conditions are satisfied if this variety is directly representable by a finite set of finite algebras. One task of the paper is to show that virtually no lattice varieties possess definable principal congruences. However, the main purpose of the paper is to apply the new criterion in proving that every para primal variety (congruence permutable variety generated by finitely many para primal algebras) is finitely axiomatizable. The paper also contains a completely new approach to the structure theory of para primal varieties which complements and extends somewhat the recent work of Clark and Krauss.     

Definable groups of partial automorphisms

The motivation for this paper is to extend the known model-theoretic treatment of differential Galois theory to the case of linear difference equations (where the derivative is replaced by an automorphism). The model-theoretic difficulties in this case arise from the fact that the corresponding theory ACFA does not eliminate quantifiers. We therefore study groups of restricted automorphisms, preserving only part of the structure. We give conditions for such a group to be (infinitely) definable, and when these conditions are satisfied we describe the definition of the group and the action explicitly. We then examine the special case when the theory in question is obtained by enriching a stable theory with a generic automorphism. Finally, we interpret the results in the case of ACFA, and explain the connection of our construction with the algebraic theory of Picard–Vessiot extensions. The only model-theoretic background assumed is the notion of a definable set.      

Representing endomorphisms and principal congruences

For every algebraU there is an algebraU ^* with (up to isomorphism) the same endomorphism, subalgebra and congruence structure as that ofU, for which every finitely generated subalgebra and every finitely generated congruence ofU ^* is singly generated. The theorem is proved in a somewhat more general category theoretic context.Presented by R. W. Quackenbush.This author's research was supported by an OTKA grant from Hungary.This author's research was supported by NSERC, The Natural Sciences and Engineering Research Council of Canada.     

Definable groups and compact p-adic Lie groups

We formulate p-adic analogues of the o-minimal group conjecturesfrom the works of Hrushovski, Peterzil and Pillay [J. Amer.Math. Soc., to appear] and Pillay [J. Math. Log. 4 (2004) 147–162];that is, we formulate versions that are appropriate for groupsG definable in (saturated) P-minimal fields. We then restrictour attention to saturated models K of Th(_p) and Th(_{p, an}),record some elementary observations when G is defined over thestandard model _p, and then make a detailed analysis of the casewhere G = E(K) for E an elliptic curve over K. Essentially,our P-minimal conjectures hold in these contexts and, moreover,our case study of elliptic curves yields counterexamples toa more naive direct translation of the o-minimal conjectures.     

Relative principal congruences in congruence-modular quasivarieties

The problem of definability of relative principal congruences in relatively congruence modular (RCM) quasivarieties is investigated. The RCM quasivarieties are characterized in terms of parameterized families of finite sets of pairs of terms which define relative principal congruences. Received April 24, 1995; accepted in final form March 2, 1998.     

Weakly diagonal algebras and definable principal congruences

An algebra is called weakly diagonal if every subuniverse of its square contains the graph of an automorphism. We show that every variety generated by a finite algebra with no proper subalgebras has a weakly diagonal generator. The result is applied in several ways and, in particular, to show that every arithmetical affine complete variety of finite type has equationally definable principal congruences. This paper is dedicated to Walter Taylor. Received February 22, 2005; accepted in final form June 3, 2005. Work of the first author was supported by grant No. 5368 from The Estonian Science Foundation.