Carol L. Walker and coauthors

Volume 89 (1961) pages 451-460

On *N*-high subgroups of Abelian groups

John M. Irwin, Carol L. Walker and Elbert A. Walker

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Volume 90 (1962) pages 185-192

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

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Volume 14 (1963) pages 319-330

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.

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Volume 15 (1964) pages 157-160

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

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Scott Foresman and Co., Chicago (1963) pages 69-119

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 ^{ß}Ext*
are also considered.

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2(1972) 513-555

Quotient categories and rings of
quotients

Carol L. Walker and Elbert A. Walker

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Volume 94 (1966) pages 207-210.

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 = G*^{1},
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 *G*^{1}
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 *G*^{1}
if and only if *K*
is *ß*-quasi-neat
in *G*^{1}.

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Dunod, Paris (1968) pages 335-343

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.

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Springer-Verlag, Berlin (1966) pages 404-420

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

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Volume 10 (1966) pages 186-209

Relative homological algebra and
Abelian groups

Carol L. Walker

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Volume 23 (1972) pages 581-586

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

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Volume 17 (1973) pages 689-706

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.

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Local quasi-isomorphisms of torsion
free abelian groups

Carol L. Walker

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Volume 7 (1976) pages 347-359

Unique decomposition and isomorphic
refinement theorems in additive categories

Carol L. Walker and Robert B. Warfield, Jr.

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Volume 37 (1997) pages 1-23

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.

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Volume 11 (1996) pages 751-759

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.

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Volume 11 (1996) pages 751-759

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.

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Volume 5 (1997) pages 223-238

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.

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Volume 21 (1999) pages 137-155

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

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.

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Volume 14, Issue 9 (1999) pages 883-898

(Special Issue: The Mathematics of Fuzzy Sets.)

Copyright © 1999 John Wiley & Sons, Inc.

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.

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(1999) pages 101-170

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 *f*of
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.

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S. Rodabaugh and E. P. Klement, Eds.

Chapter, to appear

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.

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S. Rodabaugh and E. P. Klement, Eds.

Chapter, to appear

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

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Preprint

Powers of t-norms

Carol Walker and Elbert Walker

**Abstract:**

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

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hardy@nmsu.edu

Last modified: October 15, 1999