Fred W. Roush

Generalized fuzzy matrices

Abstract We give a systematic development of fuzzy matrix theory. Many of our results generalize to matrices over the two element Boolean algebra, over the nonnegative real numbers, over the nonnegative integers, and over the semirings, and we present these generalizations. Our first main result is that while spaces of fuzzy vectors do not have a unique basis in general they have a unique standard basis, and the cardinality of any two bases are equal. Thus concepts of row and column basis, row and column rank can be defined for fuzzy matrices. Then we study Greens equivalence classes of fuzzy matrices. New we give criteria for a fuzzy matrix to be regular and prove that the row and column rank of any regular fuzzy matrix are equal. Various inverses are also studied. In the next section, we obtain bounds for the index and period of a fuzzy matrix.

The spectra of nonnegative integer matrices via formal power series

An old problem in matrix theory is to determine the n-tuples of complex numbers which can occur as the spectrum of a matrix with nonnegative entries (see [BP94, Chapter 4] or [Min88, Chapter VII]). Authors have studied the case where the ntuple is comprised of real numbers [Bor95, Cia68, Fri78, Kel71, Per53, Sal72, Sou83, Sul49], the case where the matrices under consideration are symmetric [Fie74, JLL96], and the general problem [Joh81, LM99, LL79, Rea94, Rea96, Wuw97]. Various necessary conditions and sufficient conditions have been provided, but a complete characterization is known for real n-tuples only for n ≤ 4 [Kel71, Sul49] and for complex n-tuples only for n ≤ 3 [LL79]. Motivated by symbolic dynamics, Boyle and Handelman refocused attention on the nonzero part of the spectrum by making the following “Spectral Conjecture” [BH91, BH93] (see also [Boy93, §8] and [LM95, Chapter 11]). Below, a matrix A is primitive if all entries of A are nonnegative and for some n, all entries of A are strictly positive. Also,

Williams’s conjecture is false for reducible subshifts

We show that for two subshifts of finite type having exactly two irreducible components, strong shift equivalence is not the same as shift equiv- alence. This refutes the Williams conjecture [W] in the reducible case. The irreducible case remains an open problem. MATHEMATICS RESEARCH GROUP, ALABAMA STATE UNIVERSITY, MONTGOMERY, ALABAMA 36101 This content downloaded from 207.46.13.180 on Thu, 08 Sep 2016 04:22:09 UTC All use subject to http://about.jstor.org/terms

Homology of certain algebras defined by graphs

Abstract Let K(G) for a finite graph G with vertices v1,...,vn denote the K-algebra with generators X1,...,Xn and defining relations XiXj=XjXi if and only if vi is not connected to vj by an edge in G. We describe centralizers of monomials, show that the centralizer of a monomial is again a graph algebra, prove a unique factorization theorem for factorizations of monomials into commuting factors, compute the homology of K(G), and show that K(G) is the homology ring of a certain loop space. We also construct a K(π, 1) explicitly where π is the group with generators X1,...,Xn and defining relations XiXj=XjXi if and only if vi is not connected to vj by an edge in G.

DECIDABILITY OF THE ISOMORPHISM PROBLEM FOR STATIONARY AF-ALGEBRAS AND THE ASSOCIATED ORDERED SIMPLE DIMENSION GROUPS

The notion of isomorphism on stable AF- C^{\ast} -algebras is considered in this paper in the case when the corresponding Bratteli diagram is stationary, i.e. is associated with a single square primitive incidence matrix. A C^{\ast} -isomorphism induces an equivalence relation on these matrices, called C^{\ast} -equivalence. We show that the associated isomorphism equivalence problem is decidable, i.e. there is an algorithm that can be used to check in a finite number of steps whether two given primitive matrices are C^{\ast} -equivalent or not. Special cases of this problem will be considered in a forthcoming paper.

Inclines of algebraic structures

An incline is a generalization of a Boolean algebra or fuzzy algebra consisting of a semiring satisfying additive idempotence and the incline axiom xy + x = x, xy + y = y. The ideals in a ring or semigroup form an incline, as do the topologizing filters in a ring. We generalize the latter result to semirings, and obtain the structure of these inclines in specific cases.

Kapranov rank vs. tropical rank

We show that determining Kapranov rank of tropical matrices is not only NP-hard over any infinite field, but if solving Diophantine equations over the rational numbers is undecidable, then determining Kapranov rank over the rational numbers is also undecidable. We prove that Kapranov rank of tropical matrices is not bounded in terms of tropical rank, answering a question of Develin, Santos, and Sturmfels.

Nonmanipulability in two dimensions

Abstract We prove that continuous, anonymous, strategyproof social choice functions in two dimensions for Type I preferences are coordinatewise median functions m(x 1 ,…,x 1 ,a 1 ,…,a n−1 ) . These are efficient only if the a 1 are ±∞ in equal numbers. For three or more dimensions, efficiency is not possible.

Fuzzy flows on networks

Abstract We present a theory of fuzzy flows on networks generalizing [2] and find closed formulas giving necessary and sufficient conditions for admissible flows to exist and for the maximal admissible flow.

Special domains and nonmanipulability

Abstract We investigate domains on which a nonmanipulable, nondictatorial social choice function exists, having at least three distinct values. We do not make the assumptions of Kalai and Muller (1977). We classify all such 2-person functions on the domain which is the cyclic group Z m . We show that for any domain containing Z m , existence for 2 voters and existence for some n > 2 voters are equivalent. We show that for an n -person, onto, nonmanipulable social choice function F on Z m , F ( P 1 , P 2 ,…, P n ) ∈ { x 1 , x 2 ,…, x n } always, x i being the most preferred alternative under preference P i . We show that no domain containing the dihedral group admits such a social choice function. We show that there exists a domain on which all k -tuples are free for arbitrarily large k , for which such a social choice function does exist.