# COMMUTATOR THEORY WITHOUT JOIN-DISTRIBUTIVITY

@article{Lipparini1994COMMUTATORTW, title={COMMUTATOR THEORY WITHOUT JOIN-DISTRIBUTIVITY}, author={Paolo Lipparini}, journal={Transactions of the American Mathematical Society}, year={1994}, volume={346}, pages={177-202} }

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…

## 31 Citations

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

- Mathematics
- 1999

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 )

- Mathematics
- 2002

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

- Mathematics
- 2007

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

- Mathematics
- 2015

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

- Mathematics
- 2005

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

- MathematicsMFCS
- 2008

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

- Mathematics
- 2015

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

- MathematicsJ. Log. Comput.
- 2004

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.

### EVERY m-PERMUTABLE VARIETY SATISFIES THE CONGRUENCE IDENTITY αβh = αγh

- Mathematics
- 2008

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

- MathematicsJ. Log. Comput.
- 2010

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.

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