scholarly journals Deciding Simple Infinity Axiom Sets with One Binary Relation by Means of Superpostulates

2020 ◽  
pp. 201-217
Author(s):  
Timm Lampert ◽  
Anderson Nakano
Keyword(s):  
Binary Relation

The Formation of the Binary Structure in Early Rabbinic Literature

2018 ◽  
Author(s):  
Adi Ophir ◽  
Ishay Rosen-Zvi
Keyword(s):  
Detailed Analysis ◽  
Binary Relation ◽  
Rabbinic Literature ◽  
Binary Structure ◽  
Binary Formation

This chapter sets the stage for a detailed analysis of the rabbinic goy. It traces the consolidation of the binary relation and the exclusion of hybrid categories. It further traces the rabbinic tendency to erase intermediate categories (Samaritans; foreign slaves; God-fearers; heretics) and force them into the new binary formation. From this perspective a new reading of the conversion ceremony is also offered. First appearing in rabbinic literature, the ceremony transformed diffusive spaces of conversion into a sharp and unequivocal procedure of passage—a transitory, instant event. Instead of reading this procedure as an evidence of a permeable border between groups, as scholars tend to do, the chapter shows how it performs the very erection of this border as it regulates its crossing.

scholarly journals Hermes: a simple and efficient algorithm for building the AOC-poset of a binary relation

2014 ◽  
Vol 72 (1-2) ◽  
pp. 45-71 ◽  
Author(s):  
Anne Berry ◽  
Alain Gutierrez ◽  
Marianne Huchard ◽  
Amedeo Napoli ◽  
Alain Sigayret
Keyword(s):  
Binary Relation ◽  
Efficient Algorithm

scholarly journals Object-Free Definition of Categories

2013 ◽  
Vol 21 (3) ◽  
pp. 193-205
Author(s):  
Marco Riccardi
Keyword(s):  
Binary Relation ◽  
Category Theory ◽  
Final Part ◽  
One To One ◽  
Definition Of

Summary Category theory was formalized in Mizar with two different approaches [7], [18] that correspond to those most commonly used [16], [5]. Since there is a one-to-one correspondence between objects and identity morphisms, some authors have used an approach that does not refer to objects as elements of the theory, and are usually indicated as object-free category [1] or as arrowsonly category [16]. In this article is proposed a new definition of an object-free category, introducing the two properties: left composable and right composable, and a simplification of the notation through a symbol, a binary relation between morphisms, that indicates whether the composition is defined. In the final part we define two functions that allow to switch from the two definitions, with and without objects, and it is shown that their composition produces isomorphic categories.

scholarly journals Introduction to Formal Preference Spaces

2013 ◽  
Vol 21 (3) ◽  
pp. 223-233
Author(s):  
Eliza Niewiadomska ◽  
Adam Grabowski
Keyword(s):  
Binary Relation ◽  
Preference Relation ◽  
Consumer Preference ◽  
Consumer Theory ◽  
Mathematical Economics ◽  
Characteristic Relation ◽  
Finite Set ◽  
Preference Structures ◽  
Equal Interest

Summary In the article the formal characterization of preference spaces [1] is given. As the preference relation is one of the very basic notions of mathematical economics [9], it prepares some ground for a more thorough formalization of consumer theory (although some work has already been done - see [17]). There was an attempt to formalize similar results in Mizar, but this work seems still unfinished [18]. There are many approaches to preferences in literature. We modelled them in a rather illustrative way (similar structures were considered in [8]): 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 [2] which is more general [12]. 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 [10], 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.

scholarly journals Unification of Graphs and Relations in Mizar

2020 ◽  
Vol 28 (2) ◽  
pp. 173-186
Author(s):  
Sebastian Koch
Keyword(s):  
Binary Relation ◽  
The Other ◽  
Other Hand ◽  
System 2 ◽  
Definition Of

Summary A (di)graph without parallel edges can simply be represented by a binary relation of the vertices and on the other hand, any binary relation can be expressed as such a graph. In this article, this correspondence is formalized in the Mizar system [2], based on the formalization of graphs in [6] and relations in [11], [12]. Notably, a new definition of createGraph will be given, taking only a non empty set V and a binary relation E ⊆ V × V to create a (di)graph without parallel edges, which will provide to be very useful in future articles.

On first-order sentences without finite models

2004 ◽  
Vol 69 (2) ◽  
pp. 329-339 ◽  
Author(s):  
Marko Djordjević
Keyword(s):  
Binary Relation ◽  
Function Symbol ◽  
Complete Theory ◽  
Order Property ◽  
Unary Function ◽  
First Order ◽  
Finite Models

We will mainly be concerned with a result which refutes a stronger variant of a conjecture of Macpherson about finitely axiomatizable ω-categorical theories. Then we prove a result which implies that the ω-categorical stable pseudoplanes of Hrushovski do not have the finite submodel property.Let's call a consistent first-order sentence without finite models an axiom of infinity. Can we somehow describe the axioms of infinity? Two standard examples are:ϕ1: A first-order sentence which expresses that a binary relation < on a nonempty universe is transitive and irreflexive and that for every x there is y such that x < y.ϕ2: A first-order sentence which expresses that there is a unique x such that, (0) for every y, s(y) ≠ x (where s is a unary function symbol),and, for every x, if x does not satisfy (0) then there is a unique y such that s(y) = x.Every complete theory T such that ϕ1 ϵ T has the strict order property (as defined in [10]), since the formula x < y will have the strict order property for T. Let's say that if Ψ is an axiom of infinity and every complete theory T with Ψ ϵ T has the strict order property, then Ψ has the strict order property.Every complete theory T such that ϕ2 ϵ T is not ω-categorical. This is the case because a complete theory T without finite models is ω-categorical if and only if, for every 0 < n < ω, there are only finitely many formulas in the variables x1,…,xn, up to equivalence, in any model of T.

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