On N-high subgroups of Abelian groups
John M. Irwin, Carol L. Walker and Elbert A. Walker

Splitting properties of high subgroups
John M. Irwin, Carol L. Walker and Elbert A. Walker

Abstract:
We investigate the relation between the splitting of an Abelian group G and the splitting of high subgroups of G. Our main result states that a reduced group G splits if and only if G/G_t is reduced and some high subgroup of G splits. In a sense, this reduces the splitting problem for arbitrary groups to groups with no elements of infinite height.

High extensions of Abelian groups
David K. Harrison, John M. Irwin, Carol L. Walker, and Elbert A. Walker

Abstract:
This paper is concerned with the homological properties of high subgroups, pure-high subgroups, neat subgroups, and essential subgroups.

Properties of Ext and quasi-splitting of abelian groups
Carol L. Walker

Abstract:
We show that an Abelian group G is quasi-isomorphic to a group that splits over its torsion subgroup if and only if the short exact sequence 0 -> G_t -> G -> G/G_t -> 0 represents an element of finite order in Ext(G/G_t,G_t), where G_t denotes the torsion subgroup of G. This leads to a negative answer to the question, posed by L. Fuchs at the New Mexico State University Symposium on Abelian Groups, June 1962: If a group G is quasi-isomorphic to a group that splits oer its torsion subgroup, does G necessarily split over its torsion subgroup?

On p^ß-pure sequences of Abelian groups
John M. Irwin, Carol L. Walker and Elbert A. Walker

Abstract:
This paper is an investigation of a generalization of purity that arose homologically. One of the central considerations is a discussion of the short exact sequences which represent the elements of p-height ß in Ext(A,B), where p is any prime and ß is any ordinal number. Injectives and projectives for p^ßExt are also considered.

Quotient categories and rings of quotients
Carol L. Walker and Elbert A. Walker

On a certain purification problem for primary Abelian groups
Fred Richman and Carol L. Walker

Abstract:
The general purification problem is to ascertain precisely which subgroups of a subgroup A of an abelian p-group G are the intersections of A with a pure subgroup of G. This note solves the purification problem for A = G¹, the subgroup of elements having infinite height in G. If ß is a cardinal number, we say that K is ß-quasi-neat in G if the quotient group obtained by the intersection of pG with K modulo the subgroup pK contains no more than ß elements. The main theorem states: Let G be an abelian p-group, K a subgroup of G¹ and the final rank of a high subgroup of G. There exists a pure subgroup P of G such that K is the intersection of P with G¹ if and only if K is ß-quasi-neat in G¹.

Projective classes of Abelian groups
Fred Richman, Carol L. Walker, Elbert A. Walker

Abstract:
Associated with any class of objects of an Abelian category is the class of proper (with respect to that class), short exact sequences. Also, given any class of short exact sequences, there is associated the class of objects that are projective with respect to that class. We give an elementary but useful criterion for the projective closure of a class of objects to be equal to the class of direct summands of direct sums of objects in the class, and describe a condition on a class of objects which implies that the projective closure of that class contains nontrivial divisible groups. Using these two theorems we are able to describe the projective closure of the class of torsion complete Abelian p-groups. Also the projective closure of the class of reduced torsion free groups is described -- it is the class of all torsion free groups. Finally, several theorems are given which relate the existence of divisible groups in the projective closure of a class to properties of the proper short exact sequences determined by that class.

Quotient categories of modules
Carol L. Walker and Elbert A. Walker

Abstract:
Let C be an Abelian category with exact direct limits and a generator. Then C has a proper generator: a generator is proper if it has the property that an inclusion map A -> U induces an isomorphism Hom_C(U,U) -> Hom_C(A,U) if and only if A = U.  The class E of rings that are endomorphism rings of proper generators are a subject of study in this paper. A ring is in E if and only if the set of right ideals I such that the natural map Hom_R(R,R) -> Hom_R(I,R) is an isomorphism is an idempotent topologizing filter. The class E contains all commutative rings and all self-injective rings.  The endomorphism ring of a proper generator U is self-injective if and only if U is injective. If U and V are proper generators of Abelian categories C and D with exact direct limits, and if U and V  have isomorphic endomorphism rings, then C and D are equivalent categories. In the category of Abelian p-groups, a complete description of all proper generators remains unknown, but it is shown that the class includes all non-reduced generators, all generators G such that Pext(G,G) = 0 and G is not torsion-complete, and all generators that are direct sums of cyclic groups.

Relative homological algebra and Abelian groups
Carol L. Walker

Projective classes of completely decomposable Abelian groups
Carol L. Walker

Abstract:
An Abelian group is completely decomposable if it has a decomposition as a direct sum of rank one groups, where the rank one groups are the group Q of rationals, the p-primary subgroups of the quotient group Q/Z of Q modulo the subgroup Z of integers, and all subgroups of these groups. Given any class of rank one groups, the projective closure of that class consists of all summands of direct sums of groups in the class and Z. The projective closure of the class of all rank one groups is the class of all completely decomposable groups. The corresponding class of proper short exact sequences for the class of all completely decomposable groups is characterized. For short exact sequences of torsion groups, it is the pure short exact sequences for which the sequence of maximum divisible subgroups is exact. Among short exact sequences of torsion-free groups, it is those sequences for which every coset contains an element of maximal height. For a sequence N = {n(p)} of non-negative integers and symbols infinity, indexed by the set of primes p, and any abelian group G, let N(G) be the intersection over all primes of the subgroups p^n(p)G. In general, a short exact sequence 0 -> A -> B -> C -> 0 is proper with respect to the class of all rank one groups if and only if for every such sequence N, the sequence 0 -> N(A) -> N(B) -> N(C) -> 0  is exact. By restricting to the relevant set of sequences, this generalizes to characterizations of the proper exact sequences that arise from different classes of rank one groups. The projective dimension of the corresponding homological algebra is determined in a few special cases.

Projective classes of cotorsion groups
Carol L. Walker

In this paper we take several classes of cotorsion groups, and describe the projective closure of the class. We show that the projective closure of the class of all reduced cotorsion groups contains enough projectives and that a group is in the projective closure of this class if and only if it is the direct sum of a free abelian group and a divisible abelian group with a direct summand of a direct sum of reduced cotorsion groups. Similar theorems are obtained for a number of other classes of cotorsion groups, including the class of all reduced p-adic cotorsion groups and the class of all reduced torsion free p-adic cotorsion groups.

Local quasi-isomorphisms of torsion free abelian groups
Carol L. Walker

Unique decomposition and isomorphic refinement theorems in additive categories
Carol L. Walker and Robert B. Warfield, Jr.

Stone algebra extensions with bounded dense sets
Mai Gehrke, Carol Walker, Elbert Walker

Abstract:
Stone algebras have been characterized by Chen and Gratzer in terms of triples (B,D,f), where D is a distributive lattice with 1, B is a Boolean algebra, and f is a bounded lattice homomorphism from B into the lattice of filters of D. If D is bounded, the construction of these characterizing triples is much simpler, since the homomorphism f can be replaced by one from B into D itself. The triple construction leads to natural embeddings of a Stone algebra into ones with bounded dense set. These embeddings correspond to a complete sublattice of the distributive lattice of lattice congruences of S. In addition, the largest embedding is a reflector to the subcategory of Stone algebras with bounded dense sets and morphisms preserving the zero of the dense set. Since this is not a full subcategory, the reflector is not idempotent.

De Morgan systems on the unit interval
Mai Gehrke, Carol Walker, Elbert Walker

Abstract:
Logical connectives on fuzzy sets arise from those on the unit interval. The basic theory of these connectives is cast in an algebraic spirit with an emphasis on equivalence between the various systems that arise. Special attention is given to DeMorgan systems with strict Archimedean t-norms and strong negations. A typical result is that any DeMorgan system with strict t-norm and strong negation is isomorphic to one whose t-norm is multiplication.

Some comments on interval-valued fuzzy sets
Mai Gehrke, Carol Walker, Elbert Walker

Abstract:
This paper presents a framework for fuzzy set theory in which fuzzy values are subintervals of the unit interval.

A mathematical setting for fuzzy logics
Mai Gehrke, Carol Walker, Elbert Walker

Abstract:
The setup of a mathematical propositional logic is given in algebraic terms, describing exactly when two choices of truth value algebras give the same logic. The propositional logic obtained when the algebra of truth values is the real numbers in the unit interval equipped with minimum, maximum and  x' = 1 - x for conjunction, disjunction and negation, respectively, is the standard propositional fuzzy logic. This is shown to be the same as three-valued logic. The propositional logic obtained when the algebra of truth values is the set of subintervals of the unit interval with component-wise operations, is propositional interval-valued fuzzy logic. This is shown to be the same as the logic given by a certain four element lattice of truth values. Since both of these logics are equivalent to ones given by finite algebras, it follows that there are finite algorithms for determining when two statements are logically equivalent within either of these logics. On this topic, normal forms are discussed for both of these logics.

A note on negations and nilpotent t-norms
Mai Gehrke, Carol Walker, Elbert Walker

In this paper, we explore general relationships among negations, convex Archimedean nilpotent t-norms, and automorphisms of the lattice consisting of the unit interval with its natural order. Each nilpotent t-norm has a (strong) negation naturally associated with it, namely, the residual of the t-norm -- the sup of the elements y whose t-norm with x is 0. The same negation is determined by the formula f ^-1(f(0)/f(x)) where fis a (multiplicative) generating function for the t-norm.

A system consisting of the unit interval with its natural order, a t-norm, a t-conorm and a decreasing unary operation * is called De Morgan if the t-norm, t-conorm and * satisfy De Morgan's laws; Stone if the t-norm of x and y is 0 if and only if y is less than or equal to x* and the t-conorm of x* with x** equals 1 ; and Boolean if it is both De Morgan and Stone. A system is shown to be Boolean if and only if * is the residual of the t-norm, and the t-norm, t-conorm and * satisfy De Morgan's laws. We also look at De Morgan, weak-Boolean and Stone systems on the lattice of subintervals of the unit interval and compare properties of related systems on the two lattices.

Averaging operators on the unit interval
Mai Gehrke, Carol Walker, Elbert Walker

Abstract
In working with negations and t-norms, it is not uncommon to call upon the arithmetic of the real numbers even though that is not part of the structure of the unit interval as a bounded lattice. To develop a self-contained system, we incorporate an averaging operator, which provides a (continuous) scaling of the unit interval that is not available from the lattice structure. The interest here is in the relations among averaging operators and t-norms, t-conorms, negations, and their generators.

Algebraic aspects of fuzzy sets and fuzzy logic
Mai Gehrke, Carol Walker, Elbert Walker

Contents: Introduction; The unit interval, t-norms, t-conorms and negations; De Morgan systems with strict t-norms; De Morgan systems with nilpotent t-norms; Averaging operators on the unit interval; Interval-valued fuzzy sets; A mathematical setting for fuzzy logics.

Abstract:
This paper is expository, mainly a survey of some of our work on the algebraic systems that arise in fuzzy set theory and logic. Our point of view is the algebraic one: when are the various systems that arise isomorphic, and what are their symmetries or automorphisms? The bulk of the material centers around t-norms, and a typical concern is with the unit interval endowed with its natural order structure, a t-norm, and a negation. A fundamental problem is to determine when two such algebraic systems are isomorphic. We begin with some basic facts about the unit interval with its order structure. This includes some information about various groups that will play a role throughout. We introduce t-norms, gives representation theorems for them, and determine their isomorphy and their automorphism groups. We do the same for negations.

We introduce systems, called De Morgan systems, that concern the unit interval with its usual order, a t-norm, a negation, and the corresponding t-conorm. A typical result is that any De Morgan system with strict t-norm and strong negation is isomorphic to one whose t-norm is multiplication. The non-uniqueness of the negation in a strict De Morgan system is discussed. The generators of strict t-norms are determined up to elements of the multiplicative group of positive real numbers. This group can be viewed as a subgroup of the automorphism group of the unit interval, and its normalizer in that group yields a special set of t-norms. We determine that normalizer and give explicit formulas for the resulting t-norms, t-conorms and negations.

Another typical result is that any De Morgan system with nilpotent t-norm and strong negation is isomorphic to one whose t-norm is the well-known Lukasiewicz t-norm. This provides a unique representation of nilpotent t-norms by elements of the automorphism group of the unit interval, which puts a (nearly) natural group structure on the set of nilpotent t-norms and also leads to theorems similar to those for strict t-norms. We explore general relationships among negations, convex Archimedean nilpotent t-norms, and automorphisms of the unit interval. Each nilpotent t-norm has a (strong) negation naturally associated with it. The notions of Stone and Boolean systems are introduced. A deMorgan system is shown to be a Boolean system if and only if the t-norm is nilpotent and the negation is the one naturally associated with it.

We consider averaging operators -- binary operations on the unit interval that are commutative; strictly increasing in each variable; convex (continuous); idempotent; and bisymmetric. All averaging operators are isomorphic to the arithmetic mean via an automorphism fof the unit interval (a generator for the averaging operator) that takes the given average of two elements x and y to the arithmetic mean of f(x) and f(y). Averaging operators provide a (continuous) scaling of the unit interval that is not provided by the lattice structure. We consider mean systems which consist of the unit interval with its natural order together with an averaging operator, and note that these algebras have no nontrivial automorphisms. We show that each averaging operator on the unit interval naturally defines a negation by the equation: the average of an element with its negation equals the average of 0 and 1, and show that the averaging operator is "self-dual'' with respect to this negation. We relate an averaging operator to the nilpotent t-norms that determine the same negation and find a natural one-to-one correspondence between averaging operators and nilpotent t-norms. Corresponding averaging operators and nilpotent t-norms determining the same negation. This correspondence relates the Lukasiewicz t-norm to the arithmetic mean, both of which lead to the standard negation 1 - x, for example. We consider what happens in the general case. We consider De Morgan systems with averaging operators and generalize the families of Frank t-norms and nearly Frank t-norms in this setting.

We develop the basic theory of t-norms, negations, and t-conorms on interval-valued fuzzy sets, where the unit interval is replaced by the lattice of subintervals of the unit interval. There is a view that models based on the unit interval are inadequate, that assigning an exact number to an expert's opinion is too restrictive, and that the assignment of an interval of values is more realistic. The basic theory goes through, and we develop that theory. We also look at De Morgan, weak-Boolean and Stone systems on the lattice of subintervals\ and compare properties of related systems with those on the unit interval.

The setup of a mathematical propositional logic is given in algebraic terms, describing exactly when two choices of truth value algebras give the same logic. The propositional logic obtained when the algebra of truth values is the real numbers in the unit interval equipped with minimum, maximum and 1 - x for conjunction, disjunction and negation, respectively, is the standard propositional fuzzy logic. This is shown to be the same as three-valued logic. The propositional logic obtained when the algebra of truth values is the set of subintervals of the unit interval with component-wise operations, is propositional interval-valued fuzzy logic. This is shown to be the same as the logic given by a certain four element lattice of truth values. Since both of these logics are equivalent to ones given by finite algebras, it follows that there are finite algorithms for determining when two statements are logically equivalent within either of these logics. On this topic, normal forms are described for both of these logics.

Varieties generated by De Morgan systems
Mai Gehrke, Carol Walker, Elbert Walker

Abstract:
The propositional logics determined by two De Morgan systems with strict t-norms are equal if and only if the De Morgan systems are isomorphic. Otherwise, they are incomparable.

Groups and triple systems
Carol Walker and Elbert Walker

Abstract:
Continuous archimedean t-norms are generated by automorphisms the unit interval with its usual order structure. Such strict t-norms are given by x* y = f ^-1(f(x)f(y)) and nilpotent ones by x * y = f ^-1(max{(f(x)+f(y)-1),0}). Let A be the group of automorphisms of the unit interval with its natural order, the group operation being composition of functions. Certain subgroups of A play an important role in the theory, for example the multiplicative group of positive real numbers, which is embedded in A by r(x) = x^r. Some standard families of t-norms are in natural one-to-one correspondence with subgroups of A. We examine this phenomenon, and various other group theoretic aspects of t-norm theory.

Powers of t-norms
Carol Walker and Elbert Walker

Abstract:
The rth power of a continuous t-norm is defined for positive real numbers r, generalizing the notion of the diagonal (2nd power). The increasing functions from the unit interval to itself that are realized as the rth power of some continuous t-norm are identified.