Linear forms and axioms of choice
نویسنده
چکیده
We work in set-theory without choice ZF. Given a commutative field K, we consider the statement D(K): “On every non null K-vector space there exists a non-null linear form.” We investigate various statements which are equivalent to D(K) in ZF. Denoting by Z2 the two-element field, we deduce that D(Z2) implies the axiom of choice for pairs. We also deduce that D(Q) implies the axiom of choice for linearly ordered sets isomorphic with Z.
منابع مشابه
On characterizations of the fully rational fuzzy choice functions
In the present paper, we introduce the fuzzy Nehring axiom, fuzzy Sen axiom and weaker form of the weak fuzzycongruence axiom. We establish interrelations between these axioms and their relation with fuzzy Chernoff axiom. Weexpress full rationality of a fuzzy choice function using these axioms along with the fuzzy Chernoff axiom.
متن کاملConsistent Probabilistic Social Choice
Two fundamental axioms in social choice theory are consistency with respect to a variable electorate and consistency with respect to components of similar alternatives. In the context of traditional non-probabilistic social choice, these axioms are incompatible with each other. We show that in the context of probabilistic social choice, these axioms uniquely characterize a function proposed by ...
متن کاملMaking choices with a binary relation:
This article presents an axiomatic analysis of the best choice decision problem from a reflexive crisp binary relation on a finite set (a digraph). With respect to a transitive digraph, optimality and maximality are usually accepted as the best fitted choice axioms to the intuitive notion of best choice. However, beyond transitivity (resp. acyclicity), optimality and maximality can characterise...
متن کاملBASES AND CIRCUITS OF FUZZIFYING MATROIDS
In this paper, as an application of fuzzy matroids, the fuzzifying greedy algorithm is proposed and an achievableexample is given. Basis axioms and circuit axioms of fuzzifying matroids, which are the semantic extension for thebasis axioms and circuit axioms of crisp matroids respectively, are presented. It is proved that a fuzzifying matroidis equivalent to a mapping which satisfies the basis ...
متن کاملBASE AXIOMS AND SUBBASE AXIOMS IN M-FUZZIFYING CONVEX SPACES
Based on a completely distributive lattice $M$, base axioms and subbase axioms are introduced in $M$-fuzzifying convex spaces. It is shown that a mapping $mathscr{B}$ (resp. $varphi$) with the base axioms (resp. subbase axioms) can induce a unique $M$-fuzzifying convex structure with $mathscr{B}$ (resp. $varphi$) as its base (resp. subbase). As applications, it is proved that bases and subbase...
متن کامل