Congruence Axioms Research Articles

Side-side-angle with fixed angles and the cyclic quadrilateral

Among the triangle congruence axioms, the side-side-angle (SsA) axiom states that two triangles are congruent if and only if two pairs of corresponding sides and the angles opposite the longer sides are equal. The modification of the SsA axiom provides a construction with two triangle sequences . We require that the opposite angles of the equivalent shorter sides be fixed and the longer sides be equal. The locus of the intersection points of other sides of triangles is derived to be a hyperbola, and in a generalized form defined by a complete quadrilateral, it is a conic section.

Side-side-angle triangle congruence axiom and the complete quadrilaterals

<abstract><p>Among the triangle congruence axioms, the side-side-angle (SsA) axiom states that two triangles are congruent if and only if two pairs of corresponding sides and the angles opposite the longer sides are equal. We construct two triangle sequences in which the items satisfy a modified condition. We require that the opposite angles of the shorter sides be equal. The locus of the intersection points of other sides of triangles is derived to be a hyperbola, and in a generalized form defined by a complete quadrilateral, it is a conic section.</p></abstract>

The Geometry of Otto Selz’s Natural Space

Following ideas elaborated by Hering (Grundzuge der Lehre vom Lichtsinn, Springer, Berlin, 1920) in his celebrated analysis of color, the psychologist and gestalt theorist Otto Selz developed in the 1930s a theory of “natural space”, i.e., space as it is conceived by us. Selz’s thesis is that the geometric laws of natural space describe how the points of this space are related to each other by directions which are ordered in the same way as the points on a sphere. At the end of one of his articles, Selz (Zeitschrift fur Psychologie 114:351–362, 1930a, p. 358ff) tries to derive within his framework two of Hilbert’s axioms for Euclidean geometry. Such derivations (if successful) would, according to Selz, disclose the psychological origin and meaning of the geometric axioms and would thus contribute to a clarification of their epistemological status. In the present article Selz’s theory is explained and analyzed, his basic principles are amended, and implicit assumptions are made explicit. It is shown that the resulting system is one of ordered affine geometry in which all of Hilbert’s axioms except the axioms of congruence and those of continuity are derivable. The primary aim of the present paper is to make explicit the basic principles behind Selz’s geometry of natural space. The question of the logical independence of these principle is not investigated.

Rational Choice with Intransitive Preferences

Traditional rational choice theory assumes that the weak preference relation of an agent is an ordering that is it satisfies reflexivity, completeness and transitivity. It is also well known that the ordering property is essential to build the traditional ordinal utility analysis of consumer behaviour. However, there can be many situations when the weak preference relation of an agent may violate transitivity property, and hence, is not an ordering. In such situations traditional ordinal utility analysis breaks down. This paper develops a framework and discusses all the important results of rational choice theory when preferences are intransitive. It looks at weaker rationality properties such as quasi-transitivity and acyclicity and based on that it introduces weaker concepts of rationality such as quasi-transitive rationality and acyclic rationality and characterizes them. It also brings in the congruence axioms and property of path independence, and establishes the link with rationality. Finally, it analyzes how the results will change if we bring in restricted domain assumption of the choice function. JEL Classification: D01, D10, D11

Transitive and acyclic rationality indicators of fuzzy choice functions on base domain

In the present paper we introduce the indicators of the fuzzy transitive congruence axiom, fuzzy direct-revelation axiom and fuzzy acyclic congruence axiom. These indicators measure the degree to which a fuzzy choice function satisfies these axioms. We use the indicators of fuzzy transitive congruence axiom and fuzzy acyclic congruence axiom to calculate the minimum degree to which the direct fuzzy revealed preference relation is the transitive and acyclic respectively. We established that (i) the degree to which the fuzzy choice function is full rational is the degree to which it satisfies fuzzy transitive congruence axiom and (ii) the degree to which the fuzzy choice function is acyclic rational is the minimum degree to which it satisfies fuzzy direct-revelation axiom and its fuzzy revealed preference is acyclic. We show that a similarity relation on the set of fuzzy choice functions preserves the indicators of fuzzy transitive congruence axiom, fuzzy direct-revelation axiom, fuzzy acyclic congruence axiom and (transitive and acyclic) rationality indicators.

ON INTERRELATIONS BETWEEN FUZZY CONGRUENCE AXIOMS

In this paper we establish interrelations between fuzzy direct revelation axiom (FDRA), fuzzy transitive-closure coherence axiom (FTCCA), fuzzy consistent-closure coherence axiom (FCCCA) and fuzzy intermediate congruence axiom (FICA). We also establish their relationships with weak fuzzy congruence axiom (WFCA), strong fuzzy congruence axiom (SFCA) and weak axiom of fuzzy revealed preference (WAFRP). Condition for equivalence of fuzzy Arrow axiom (FAA) and weak fuzzy congruence axiom (WFCA) on arbitrary domain is also given.

A further study on rationality conditions of fuzzy choice functions

The investigation of rationality plays a central role in the study of fuzzy choice functions. In this paper, we deal with some important rationality conditions of fuzzy choice functions, including Weak (Strong) Fuzzy Congruence Axiom, Weak (Strong) Axiom of Fuzzy Revealed Preference, fuzzy versions of famous crisp conditions α , β , γ and δ etc. in the framework of the Banerjee choice function. We systematically investigate the relationships between these conditions under the assumption that every involved choice set is normal and verify some equivalence results for arbitrary t-norms. As a result, a fuzzy version of the Arrow–Sen theorem is presented. The research makes a significant difference compared with the current investigation in the framework of the Georgescu choice function.

A System of Axioms for Hyperbolic Geometry

Three–dimensional hyperbolic geometry is characterized using axioms of order, incidence, dimension, continuity and, instead of an axiom of parallels, there is an axiom of “rigidity” and, rather than several axioms of congruence, there is one axiom of symmetry. It is claimed that this system of axioms is simpler than the system of independent axioms of Moore [15].

Nash rationalization of collective choice over lotteries

To test the joint hypothesis that players in a noncooperative game (allowing mixtures over pure strategies) consult an independent preference relation and select a Nash equilibrium, it suffices to study the reaction of the revealed collective choice upon changes in the space of strategies available to the players. The joint hypothesis is supported if the revealed choices satisfy an extended version of Richter’s congruence axiom together with a contraction–expansion axiom that models the noncooperative behavior. In addition, we provide sufficient and necessary conditions for a binary relation to have an independent ordering extension, and for individual choices over lotteries to be rationalizable by an independent preference relation.

Congruence indicators for fuzzy choice functions

We introduce the congruence indicators WFCA(·) and SFCA(·) corresponding to fuzzy congruence axioms WFCA and SFCA. These indicators measure the degree to which a fuzzy choice function verifies the axioms WFCA and SFCA, respectively. The main result of the paper establishes for a given choice function the relationship between its congruence indicators and some rationality conditions. One obtains a fuzzy counterpart of the well-known Arrow–Sen theorem in crisp choice functions theory.

Consistency indicators for fuzzy choice functions

For a fuzzy choice function C we study two consistency indicators F α ( C), F β ( C) and a congruence indicator WFCA( C). The indicators F α ( C) and F β ( C) express the degree to which C verifies the extension to the fuzzy case of Sen's conditions α and β, while WFCA( C) evaluates the degree to which C verifies the weak fuzzy congruence axiom WFCA. The main result of the paper shows that WFCA( C) is the minimum of F α ( C) and F β ( C). The three indicators can be used for obtaining a hierarchy of a class of fuzzy choice functions.

Arrow’s Axiom and Full Rationality for Fuzzy Choice Functions

A classical result for crisp choice functions shows the equivalence between Arrow axiom and the property of full rationality. In this paper we study a fuzzy form of Arrow axiom formulated in terms of the subsethood degree and of the degree of equality (of fuzzy sets). We prove that a fuzzy choice function satisfies Fuzzy Arrow Axiom if and only if it is (fuzzy) full rational. We also show that these conditions are also equivalent with weak and strong fuzzy congruence axioms WFCA and SFCA. It is studied the Arrow index, a new concept that indicates the degree to which a fuzzy choice function satisfies the Fuzzy Arrow Axiom.

Degree of dominance and congruence axioms for fuzzy choice functions

In this paper we introduce the degree of dominance of an alternative x with respect to an available fuzzy set of alternatives. Interpreting an available fuzzy set of alternatives as a criterion in decision making the degree of dominance establishes a hierarchy of alternatives with respect to this criterion. With the degree of dominance new congruence axioms for fuzzy choice functions are formulated.

Revealed Preference, Congruence and Rationality: A Fuzzy Approach

In a previous paper connections between the congruence axioms WFCA, SFCA and the revealed preference axioms WAFRP, SAFRP for fuzzy choice functions whose domain contains the characteristic function...

A New Congruence Axiom and Transitive Rational Choice

Rationality in choice theory has been an abiding concern of decision theorists. A rationality postulate of considerable significance in the literature is the weak congruence axiom of Richter and Sen. It is well known that in discrete choice contexts of the classical type (i.e. all non‐empty finite subsets of a given set comprise the set of choice problems) this axiom is equivalent to full rationality. The question is: will a weakening of the weak congruence axiom suffice to imply full rationality? This is the question we take up in this paper. We propose a weaker new congruence axiom which along with the Chernoff axiom implies full rationality. The two axioms are independent. We also study interesting properties of these axioms and their interconnections through examples.

Internal Languages for Autonomous and ∗-Autonomous Categories

We introduce a family of type theories as internal languages for autonomous (or symmetric monoidal closed) and ∗-autonomous categories, in the same sense that the simply-typed lambda-calculus with surjective pairing is the internal language for cartesian closed categories. The rules for the typing judgements are presented in the style of Gentzen's Sequent Calculus. A notable feature is the systematic treatment of naturality conditions by expressing the categorical composition, or cut in the type theory, by explicit substitution. We use let-constructs, one for each of the three type constructors tensor unit, tensor and linear function space, and a Parigot-style mu-abstraction to express the involutive negation. We show that the eight equality and three commutation congruence axioms of the ∗-autonomous type theory characterise ∗-autonomous categories exactly. More precisely we prove that there is a canonical interpretation of the (∗-autonomous) type theories in ∗-autonomous categories which is complete i.e. for any type theory, there is a model (i.e. ∗-autonomous category) whose theory is exactly that. The associated rewrite systems are all strongly normalising; modulo a simple notion of congruence, they are also confluent. As a corollary, we solve a Coherence Problem a la Lambek: the equality of maps in any ∗-autonomous category freely generated from a discrete graph is decidable.

Multi-valued demand and rational choice in the two-commodity case

We show that the weak and strong congruence axioms for consumer demand correspondences are equivalent if there are only two commodities. This result generalizes a well-known analogous equivalence theorem that applies to single-valued demands.

Ebenen mit Kongruenz

Euclidean planes are characterized as affine planes with a congruence relation on the set of the pairs of points. Furthermore we study the consequences of the congruence axioms for other incidence structures e.g. finite or half ordered ones, and discuss the question when the 3-reflection theorem is valid.

Analoga der steinerschen konstruktion in einer absoluten geometrie

We consider an absolute geometry with the following base of axioms: Hilbert's plane axioms of incidence, order and congruence and a circle axiom. Thus no parallelism and not much continuity is involved. In this geometry the metric cannot be determined by Steiner's basic structure “fixed circle with centre”. In this work it will be proved that the following basic figures are suitable for such an absolute geometry in the sense that, after tracing any one of them, all constructions of second order can be done only with a ruler: 1) Two non-concentric circles, one of them with centre. 2) A unit-turner and a non-concentric circle without centre. 3) A circle with centreO and a line segmentA B with midpointM, the linesA B andO M being not orthogonal. 4) A circle with centre and two orthogonal lines, none of them passing through the centre. 5) A circle with centre and a distance-line (with their two branches).

On J. Bolyai's parallel construction

Given a model of Hilbert's incidence, order and congruence axioms (an H-plane) in which the Line-Circle Principle and the hypothesis of the acute angle hold, J. Bolyai's parallel construction is shown to yield the two parallels through the given point that have a “common perpendicular at infinity” with the given line through the ideal points at which they meet the given line; these need not be asymptotic parallels, for the plane need not be hyperbolic. Two new characterizations of hyperbolic planes are given, based on W. Pejas' classification of H-planes. The role of Archimedes' axiom is clarified.