author={Paolo Lipparini},
  journal={Transactions of the American Mathematical Society},
  • P. Lipparini
  • Published 1994
  • Mathematics
  • Transactions of the American Mathematical Society
We develop Commutator Theory for congruences of general alge- braic systems (henceforth called algebras) assuming only the existence of a ternary term d such that d(a, b, b)(a, a)a(a, a)d(b, b, a), whenever a is a congruence and aab . Our results apply in particular to congruence modular and n-permutable varieties, to most locally finite varieties, and to inverse semigroups. We obtain results concerning permutability of congruences, abelian and solv- able congruences, connections between… 

Congruence Modularity Implies the Arguesian Law for Single Algebras with a Difference Term

Abstract Recently, a generalization of commutator theory has been developed for algebraic systems belonging to a congruence modular variety. This general commutator theory is used here both to

The Lattice of Lambda Theories ( Regular Research Paper )

The lattice Ì of lambda theories is isomorphic to the congruence lattice of the term algebra of the minimal lambda theory ¬. This remark is the starting point for studying the structure of Ì by

Logic Colloquium 2004: Tolerance intersection properties and subalgebras of squares

Tolerance identities can be used [5] in order to provide a fairly simple proof of a classical result by R. Freese and B. Jonsson asserting that every congruence modular variety is in fact Arguesian.

A finite basis theorem for difference-term varieties with a finite residual bound

We prove that if V is a variety of algebras (i.e., an equationally axiomatizable class of algebraic structures) in a finite language, V has a difference term, and V has a finite residual bound, then

Optimal Mal’tsev conditions for congruence modular varieties

Abstract.For varieties, congruence modularity is equivalent to the tolerance intersection property, TIP in short. Based on TIP, it was proved in [5] that for an arbitrary lattice identity implying

From lambda-Calculus to Universal Algebra and Back

The class of Church algebras is introduced to model the if-then-else instruction of programming to prove that any Church algebra with an "easy set" of cardinalitynadmits (at the top) a lattice interval of congruencesisomorphic to the free Boolean algebra with ngenerators.

Commutator Studies in Pursuit of Finite Basis Results

Several new results of a general algebraic scope are developed in an effort to build tools for use in finite basis proofs. Many recent finite basis theorems have involved assumption of a finite

The Lattice of Lambda Theories

It is shown that nontrivial quasi-identities in the language of lattices hold in the lattice of lambda theories, while every nontrivials lattice identity fails in theattachment if thelanguage of lambda calculus is enriched by a suitable finite number of constants.


It is known that congruence lattices of algebras in m-permutable varieties satisfy non-trivial identities; however, the identities discovered so far are rather artificial and seem to have little

Applying Universal Algebra to Lambda Calculus

It is shown that lambda calculus and combinatory logic satisfy interesting algebraic properties and the indecomposable semantics is equationally incomplete, and this incompleteness is as wide as possible.



A Characterization of Identities Implying Congruence Modularity I

In his thesis and [24], J. B. Nation showed the existence of certain lattice identities, strictly weaker than the modular law, such that if all the congruence lattices of a variety of algebras

Commutator theory for congruence modular varieties

Introduction In the theory of groups, the important concepts of Abelian group, solvable group, nilpotent group, the center of a group and centraliz-ers, are all defined from the binary operation [x,


We prove that if V is any infinite-dimensional vector space over any uncountable field F, then the congruence lattice (=subspace lattice) of V cannot be represented as a congruence lattice (of any

Three remarks on the modular commutator

First a problem of Ralph McKenzie is answered by proving that in a finitely directly representable variety every directly indecomposable algebra must be finite. Then we show that there is no local

Some characterizations of the commutator

We start with a characterization of the modular commutator that was given by E. Kiss and the first author in [DK] and explore some of its consequences. Implicit in this characterization is that both

Some applications of the term condition

In our main result we describe a countable algebraic lattice L which is not isomorphic to the congruence lattice Con S of any semigroup S. B. J6nsson asked in (1975) whether there was any algebraic

An Order-Theoretic Property of the Commutator

It is shown that any solvable E-minimal algebra is leftnilpotent, any finite algebra whose congruence lattice contains a 0, 1-sublattice isomorphic to M3 is left nilpotent and any homomorphic image of a finite abeliangebra is left and right nilpotents.

M n as a 0, 1-Sublattice of Con A Does not Force the Term Condition

For every n > 3 there exists a finite nonabelian algebra whose congruence lattice has Mn as a 0, 1-sublattice. This answers a question of R. McKenzie and D. Hobby. DEFINITION 0.1. Suppose L and L1


The main result of this paper is that the class of congruence lattices of semilattices satisfies no nontrivial lattice identities. It is also shown that the class of subalgebra lattices of

Inverse semigroups

In this note it is shown that if S is a free inverse semigroup of rank at least two and if e, f are idempotents of S such that e > f then S can be embedded in the partial semigroup eSe\fSf. The proof