Generation and Characterization of Fuzzy T-preorders

This article studies T-preorders that can be generated in a natural way by a single fuzzy subset. These T-preorders are called one-dimensional and are of great importance, because every T-preorder can be generated by combining one-dimensional T-preorders.; In this article, the relation between fuzzy subsets generating the same T-preorder is given, and one-dimensional T-preorders are characterized in two different ways: They generate linear crisp orderings on X and they satisfy a Sincov-like functional equation. This last characterization is used to approximate a given T-preorder by a one-dimensional one by relating the issue to Saaty matrices used in the Analytical Hierarchical Process. Finally, strong complete T-preorders, important in decision-making problems, are also characterized.Peer ReviewedPostprint (author’s final draft

Representation of strongly independent preorders by sets of scalar-valued functions

We provide conditions under which an incomplete strongly independent preorder on a convex set X can be represented by a set of mixture preserving real-valued functions. We allow X to be infinite dimensional. The main continuity condition we focus on is mixture continuity. This is sufficient for such a representation provided X has countable dimension or satisfies a condition that we call Polarization

On expenditure functions

In this paper we present complete characterizations of the expenditure function for both utility representations and preference structures. Building upon these results, we also establish under minimal assumptions duality theorems for exıpenditure functions and utility representations, and for expenditure functions and preference structures. These results generalize previous work in this area; moreover, in the case of preferences structures they apply to non-completeı preorders

Continuity and completeness of strongly independent preorders

A strongly independent preorder on a possibly infinite dimensional convex set that satisfies two of the following conditions must satisfy the third: (i) the Archimedean continuity condition; (ii) mixture continuity; and (iii) comparability under the preorder is an equivalence relation. In addition, if the preorder is nontrivial (has nonempty asymmetric part) and satisfies two of the following conditions, it must satisfy the third: (i') a modest strengthening of the Archimedean condition; (ii') mixture continuity; and (iii') completeness. Applications to decision making under conditions of risk and uncertainty are provided

Lower Approximations by Fuzzy Consequence Operators

Peer ReviewedPostprint (author's final draft

Finite-dimensional algebras with smallest resolutions of simple modules

Let $X$ be a finitely generated left module over a left artinian ring $R$, and let $p(X)=\{l_i\}$ be the infinite sequence of nonnegative integers where $l_i$ is the length of the $i$-th term of the minimal projective resolution of $X$. We introduce a preorder relation $\le$ on the set $\{p(X)\}$ and characterize the elementary finite-dimensional algebras $\Lambda$ with the following property. Let $S$ be a simple $\Lambda$-module, and let $T$ be a finitely generated module over an arbitrary left artinian ring $R$. If the projective dimension of $S$ does not exceed the projective dimension of $T$, then $p(S)\le p(T)$. We characterize the indicated algebras by quivers with relations.Comment: Minor revisions, to appear in Journal of Algebr

On expenditure functions.

In this paper we present complete characterizations of the expenditure function for both utility representations and preference structures. Building upon these results, we also establish under minimal assumptions duality theorems for exıpenditure functions and utility representations, and for expenditure functions and preference structures. These results generalize previous work in this area; moreover, in the case of preferences structures they apply to non-completeı preorders.Expenditure functions; Utility representations; Duality; Non-complete preorders;