Principal ideals in the semigroup of binary relations on a finite set: What happens when one element is added to the set

1992 ◽  
Vol 44 (1) ◽  
pp. 129-132 ◽  
Author(s):  
Michael Breen
Keyword(s):  
Binary Relations ◽  
Principal Ideals ◽  
Finite Set

scholarly journals Finite representability of semigroups with demonic refinement

2021 ◽  
Vol 82 (2) ◽  
Author(s):  
Robin Hirsch ◽  
Jaš Šemrl
Keyword(s):  
Partial Order ◽  
Turing Machines ◽  
Binary Relations ◽  
Finite Representation ◽  
First Order ◽  
Finite Structures ◽  
Finite Base ◽  
Finite Set ◽  
Finite Representability ◽  
Representation Class

AbstractThe motivation for using demonic calculus for binary relations stems from the behaviour of demonic turing machines, when modelled relationally. Relational composition (; ) models sequential runs of two programs and demonic refinement ($$\sqsubseteq $$ ⊑ ) arises from the partial order given by modeling demonic choice ($$\sqcup $$ ⊔ ) of programs (see below for the formal relational definitions). We prove that the class $$R(\sqsubseteq , ;)$$ R ( ⊑ , ; ) of abstract $$(\le , \circ )$$ ( ≤ , ∘ ) structures isomorphic to a set of binary relations ordered by demonic refinement with composition cannot be axiomatised by any finite set of first-order $$(\le , \circ )$$ ( ≤ , ∘ ) formulas. We provide a fairly simple, infinite, recursive axiomatisation that defines $$R(\sqsubseteq , ;)$$ R ( ⊑ , ; ) . We prove that a finite representable $$(\le , \circ )$$ ( ≤ , ∘ ) structure has a representation over a finite base. This appears to be the first example of a signature for binary relations with composition where the representation class is non-finitely axiomatisable, but where the finite representation property holds for finite structures.

scholarly journals On idempotent binary relations on a finite set

1970 ◽  
Vol 20 (4) ◽  
pp. 696-702 ◽  
Author(s):  
Štefan Schwarz
Keyword(s):  
Binary Relations ◽  
Finite Set

SINGLE PEAKED FUZZY PREFERENCES IN ONE-DIMENSIONAL MODELS: DOES BLACK'S MEDIAN VOTER THEOREM HOLD?

2010 ◽  
Vol 06 (01) ◽  
pp. 1-16 ◽  
Author(s):  
JOHN N. MORDESON ◽  
LANCE NIELSEN ◽  
TERRY D. CLARK
Keyword(s):  
Median Voter ◽  
Ideal Point ◽  
Binary Relations ◽  
One Dimensional ◽  
Median Voter Theorem ◽  
Political Actors ◽  
Fuzzy Preferences ◽  
Median Position ◽  
Finite Set ◽  
Dimensional Models

Black's Median Voter Theorem is among the more useful mathematical tools available to political scientists for predicting choices of political actors based on their preferences over a finite set of alternatives within an institutional or constitutional setting. If the alternatives can be placed on a single-dimensional continuum such that the preferences of all players descend monotonically from their ideal point, then the outcome will be the alternative at the median position. We demonstrate that the Median Voter Theorem holds for fuzzy preferences. Our approach considers the degree to which players prefer options in binary relations.

scholarly journals Generation of diagonal acts of some semigroups of transformations and relations

2005 ◽  
Vol 72 (1) ◽  
pp. 139-146 ◽  
Author(s):  
Peter Gallagher ◽  
Nik Ruškuc
Keyword(s):  
Symmetric Group ◽  
Binary Relations ◽  
Finitely Generated ◽  
Finite Generation ◽  
Finite Set ◽  
Infinite Set

The diagonal right (respectively, left) act of a semigroup S is the set S × S on which S acts via (x, y) s = (xs, ys) (respectively, s (x, y) = (sx, sy)); the same set with both actions is the diagonal bi-act. The diagonal right (respectively, left, bi-) act is said to be finitely generated if there is a finite set A ⊆ S × S such that S × S = AS1 (respectively, S × S = S1A, S × S = SlASl).In this paper we consider the question of finite generation for diagonal acts of certain infinite semigroups of transformations and relations. We show that the semi-groups of full transformations, partial transformations and binary relations on an infinite set each have cyclic diagonal right and left acts. The semigroup of full finite-to-one transformations on an infinite set has a cyclic diagonal right act but its diagonal left act is not finitely generated. The semigroup of partial injections on an infinite set has neither finitely generated diagonal right nor left act, but has a cyclic diagonal bi-act. The semigroup of bijections (symmetric group) on an infinite set does not have any finitely generated diagonal acts.

AGGREGATING TRANSITIVE FUZZY BINARY RELATIONS

1995 ◽  
Vol 03 (01) ◽  
pp. 47-55 ◽  
Author(s):  
SERGEI OVCHINNIKOV
Keyword(s):  
Binary Relation ◽  
Arithmetic Mean ◽  
Geometric Characterization ◽  
Binary Relations ◽  
Aggregation Procedure ◽  
Aggregation Problem ◽  
Fuzzy Binary Relation ◽  
Finite Set

We discuss the aggregation problem for transitive fuzzy binary relations. An aggregation procedure assigns a group fuzzy binary relation to each finite set of individual binary relations. Individual and group binary relations are assumed to be transitive fuzzy binary relation with respect to a given continuous t-norm. We study a particular class of aggregation procedures given by quasi-arithmetic (Kolmogorov) means and show that these procedures are well defined if and only if the t-norm is Archimedean. We also give a geometric characterization of t-norms for which the arithmetic mean is a well-defined aggregation procedure.

scholarly journals The algebraic structure of the semigroup of binary relations on a finite set

1981 ◽  
Vol 22 (1) ◽  
pp. 57-68 ◽  
Author(s):  
Ki Hang Kim ◽  
Fred William Roush
Keyword(s):  
Algebraic Structure ◽  
Matrix Multiplication ◽  
Boolean Matrix ◽  
Ideal Structure ◽  
Binary Relations ◽  
Finite Set ◽  
Boolean Matrix Multiplication ◽  
The Ideal

In this paper we study some questions proposed by B. Schein [8] regarding the semigroup of binary relationsBxfor a finite setX: what is the ideal structure ofBx, what are the congruences onBx, what are the endomorphisms ofBx? For |X| =nit is convenient to regardBxas the semigroupBnofn×n(0, l)-matrices under Boolean matrix multiplication.

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