Introduction to Formal Preference Spaces
In the article the formal characterization of preference spaces  is given. As the preference relation is one of the very basic notions of mathematical economics , it prepares some ground for a more thorough formalization of consumer theory (although some work has already been done - see ). There was an attempt to formalize similar results in Mizar, but this work seems still unfinished . There are many approaches to preferences in literature. We modelled them in a rather illustrative way (similar structures were considered in ): either the consumer (strictly) prefers an alternative, or they are of equal interest; he/she could also have no opinion of the choice. Then our structures are based on three relations on the (arbitrary, not necessarily finite) set of alternatives. The completeness property can however also be modelled, although we rather follow  which is more general . Additionally we assume all three relations are disjoint and their set-theoretic union gives a whole universe of alternatives. We constructed some positive and negative examples of preference structures; the main aim of the article however is to give the characterization of consumer preference structures in terms of a binary relation, called characteristic relation , and to show the way the corresponding structure can be obtained only using this relation. Finally, we show the connection between tournament and total spaces and usual properties of the ordering relations.
-  Kenneth J. Arrow. Social Choice and Individual Values. Yale University Press, 1963.
-  Robert J. Aumann. Utility theory without the completeness axiom. Econometrica, 30(3): 445-462, 1962.
-  Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.
-  Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
-  Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
-  Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
-  Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.
-  Klaus E. Grue and Artur Korniłowicz. Basic operations on preordered coherent spaces. Formalized Mathematics, 15(4):213-230, 2007. doi:10.2478/v10037-007-0025-4.[Crossref]
-  Sören Halldén. On the Logic of Better. Lund: Library of Theoria, 1957.
-  Emil Panek. Podstawy ekonomii matematycznej. Uniwersytet Ekonomiczny w Poznaniu, 2005. In Polish.
-  Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441-444, 1990.
-  George F. Schumm. Transitivity, preference, and indifference. Philosophical Studies, 52: 435-437, 1987.
-  Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115-122, 1990.
-  Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25-34, 1990.
-  Wojciech A. Trybulec. Partially ordered sets. Formalized Mathematics, 1(2):313-319, 1990.
-  Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
-  Freek Wiedijk. Arrow’s impossibility theorem. Formalized Mathematics, 15(4):171-174, 2007. doi:10.2478/v10037-007-0020-9.[Crossref]
-  Krzysztof Wojszko and Artur Kuzyka. Formalization of commodity space and preference relation in Mizar. Mechanized Mathematics and Its Applications, 4:67-74, 2005.
-  Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.
-  Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.
-  Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Formalized Mathematics, 1(1):85-89, 1990.