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, y] = x −1 y −1 xy. Each of these notions, except centralizers of elements, may also be defined in terms of the commutator of normal subgroups. The commutator [M, N] (where M and N are normal subgroups of a group) is the (normal) subgroup generated by all the commutators [x, y] with x ∈ M, y ∈ N. Thus we have a binary operation in the lattice of normal subgroups. This binary operation, in combination with the lattice operations , carries much of the information about how a group is put together. The operation is also interesting in its own right. It is a com-mutative, monotone operation, completely distributive with respect to joins in the lattice. There is an operation naturally defined on the lattice of ideals of a ring, which has these properties. Namely, let [J, K] be the ideal generated by all the products jk and kj, with j ∈ J and k ∈ K. The congruity between these two contexts extends to the following facts: [M, M] is the smallest normal subgroup U of M for which M/U is a commutative group; [J, J] is the smallest ideal K of J for which the ring J/K is a commutative group; that is, a ring with trivial multiplication. Now it develops, amazingly, that a commutator can be defined rather naturally in the congruence lattices of every congruence modular variety. This operation has the same useful properties that the commutator for groups (which is a special case of it) possesses. The resulting theory has many general applications and, we feel, it is quite beautiful. In this book we present the basic theory of commutators in congruence modular varieties and some of its strongest applications. The book by H. P. Gumm [41] offers a quite different approach to the subject. Gumm developed a sustained analogy between commutator theory and affine geometry which allowed him to discover many of the basic facts about the commutator. We take a more algebraic approach, using some of the shortcuts that Taylor and others have discovered. 1 2 INTRODUCTION Historical remarks. The lattice of normal subgroups of a group, with the commutator operation, is a lattice ordered monoid. It is a residuated lattice …

Commutator theory for congruence modular varieties

A residuated lattice is an ordered algebraic structure such that is a lattice, is a monoid, and \ and / are binary operations for which the equivalences hold for all a,b,c ∈ L. It is helpful to think of the last two operations as left and right division and thus the equivalences can be seen as "dividing" on the right by b and "dividing" on the left by a. The class of all residuated lattices is denoted by ℛℒ The study of such objects originated in the context of the theory of ring ideals in the 1930s. The collection of all two-sided ideals of a ring forms a lattice upon which one can impose a natural monoid structure making this object into a residuated lattice. Such ideas were investigated by Morgan Ward and R. P. Dilworth in a series of important papers [15, 16, 45–48] and also by Krull in [33]. Since that time, there has been substantial research regarding some specific classes of residuated structures, see for example [1, 9, 26] and [38], but we believe that this is the first time that a general structural theory has been established for the class ℛℒ as a whole. In particular, we develop the notion of a normal subalgebra and show that ℛℒ is an "ideal variety" in the sense that it is an equational class in which congruences correspond to "normal" subalgebras in the same way that ring congruences correspond to ring ideals. As an application of the general theory, we produce an equational basis for the important subvariety ℛℒC that is generated by all residuated chains. In the process, we find that this subclass has some remarkable structural properties that we believe could lead to some important decomposition theorems for its finite members (along the lines of the decompositions provided in [27]).

https://my.vanderbilt.edu/constantinetsinakis/files/2014/03/final-rlpaper-final.pdf

The structure of residuated lattices

Residuation is a fundamental concept of ordered structures and categories. In this survey we consider the consequences of adding a residuated monoid operation to lattices. The resulting residuated lattices have been studied in several branches of mathematics, including the areas of lattice-ordered groups, ideal lattices of rings, linear logic and multi-valued logic. Our exposition aims to cover basic results and current developments, concentrating on the algebraic structure, the lattice of varieties, and decidability.

A Survey of Residuated Lattices

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A Survey of Abstract Algebraic Logic

Varieties of BL- algebras I: General properties.

In many familiar classes of algebraic structures kernels of congruence relations are uniquely specified by the inverse images q~-l(0)= {x [ q~(x)= 0} of a specified constant 0. On the one hand, q~-l(0) is nothing else but the 0-class of the kernel congruence of q~, on the other hand q~-~(0) can be axiomatized intrinsically, namely q l (0 ) is an ideal (in rings, Boolean algebras, or more generally in Heyt ing algebras), a normal subgroup, resp. normal subloop (in groups, resp. loops) or a filter (in Implicat ion algebras or Boolean algebras again, where 0 is replaced by the unit). In this paper we investigate common features of all the above structures by using a general notion of " ideal" , which makes sense in all universal algebras having a constant 0 and which specializes to the familar concepts of ideal, normal subgroup or filter in each of the algebras quoted above. In all universal algebras the 0-classes of congruence relations are easily seen to be ideals, but we shall require that conversely each ideal is the 0-class of a unique congruence relation. Such algebras, or ra ther classes of algebras with this proper ty will be called "classes with ideal determined congruences" or shortly ideal determined. In Part 1, after presenting the precise definitions, we shall show that the ideal determined varieties are characterized by a Mal 'cev condition, which turns out to be a combination of Fichtner 's condition for 0-regularity together with a ternary te rm r(x, y, z) which is a weakened form of Mal 'cev 's permutabil i ty term. From a result of Hagem ann it follows that ideal-determined varieties have modular congruence lattices, so the theory of commutators becomes readily available. In

H. Peter Gumm

Papers

Ideals in universal algebras