Translational hull and semigroups of binary relations

1968 ◽  
Vol 9 (1) ◽  
pp. 12-21 ◽  
Author(s):  
Mario Petrich
Keyword(s):  
Simple Semigroup ◽  
Binary Relations ◽  
Abstract Characteristic

Various semigroups of partial transformations (and more generally, semigroups of binary relations) on a set have been studied by a number of Soviet mathematicians; to mention only a few: Gluskin [2], Ljapin [4], Shutov [6], Zaretski [7], [8]. In their study the densely embedded ideal of a semigroup introduced by Ljapin [4] plays a central role. In fact, a concrete semigrou Q is described in several instances by its abstract characteristic, namely either by a set of postulates on an abstract semigroup or by a set of postulates (which are usually much simpler) on an abstract semigroup S which is a densely embedded ideal of a semigroup T isomorphic to Q. In many cases, the densely embedded ideal S is a completely 0-simple semigroup. The following theorem [3, 1.7.1] reduces the study of a semigroup Q with a weakly reductive densely embedded ideal S to the study of the translational hull of S:Theorem (Gluskin). If S is a weakly reductive densely embedded ideal of a semigroup Q, then Q is isomorphic to the translational hull ω(S) of S.

The word problem for the bifree combinatorial strict regular semigroup

1993 ◽  
Vol 113 (3) ◽  
pp. 519-533 ◽  
Author(s):  
Karl Auinger
Keyword(s):  
Regular Semigroup ◽  
Free Algebra ◽  
Simple Semigroup ◽  
Direct Products ◽  
Binary Relations ◽  
Free Object ◽  
Regular Semigroups ◽  
Binary Operations ◽  
Subdirect Products

AbstractA class of regular semigroups closed under taking direct products, regular subsemigroups and homomorphic images is ane(xistence)-variety of regular semigroups. The classof all combinatorial strict regular semigroups is thee-variety generated by the five element non-orthodox completely 0-simple semigroup and consists of all regular subdirect products of combinatorial completely 0-simple semigroups and/or rectangular bands. The bifree objecton the setXinis the natural concept of a ‘free object’ in the class.is generated by the setXand the set of formal inversesX′ under the two binary operations of multiplication · and forming the sandwich element ∧A. Henceis a homomorphic image of the absolutely free algebraof type 〈2, 2〉 generated by X ∪X′. In this paper we shall describe the associated congruence onF〈2, 2〉(X∪X′) and construct a model ofin terms of sets and binary relations. As an application, a model of the free strict pseudosemilattice on a setXis obtained.

scholarly journals The translational hull of a completely 0-simple semigroup

1968 ◽  
Vol 9 (1) ◽  
pp. 1-11 ◽  
Author(s):  
Mario Petrich
Keyword(s):  
Simple Semigroup ◽  
Linear Manifold ◽  
Ideal Extension ◽  
Partial Transformation ◽  
Transformation Semigroups ◽  
Abstract Characteristic ◽  
The Given ◽  
Partial Homomorphism

The translational hull Ω(S) of a semigroup S plays an important role in the theory of ideal extensions of semigroups. In fact, every ideal extension of S by a semigroup T with zero can be constructed using a certain partial homomorphism of T\0 into Ω(S); a particular case of interest is when S is weakly reductive (see §4.4 of [3], [2], [7]). A theorem of Gluskin [6, 1.7.1] states that if S is a weakly reductive semigroup and a densely embedded ideal of a semigroup Q, then Q and Ω(S) are isomorphic. A number of papers of Soviet mathematicians deal with the abstract characteristic (abstract semigroup, satisfying certain conditions, isomorphic to the given semigroup) of various classes of (partial) transformation semigroups in terms of densely embedded ideals (see, e.g., [4]). In many of the cases studied, the densely embedded ideal in question is a completely 0-simple semigroup, so that Gluskin's theorem mentioned above applies. This enhances the importance of the translational hull of a weakly reductive, and in particular of a completely 0-simple semigroup. Gluskin [5] applied the theory of densely embedded ideals (which are completely 0-simple semigroups) also to semigroups and rings of endomorphisms of a linear manifold and to certain classes of abstract rings.

FACTOR ANALYSIS OF GEOGRAPHIC COORDINATES AT POINTS OF THE CHANNEL OF A SMALL RIVER ON SPACE IMAGES

2020 ◽  
Author(s):  
Peter Matveevich Mazurkin ◽  
Yana Oltgovna Georgieva
Keyword(s):  
Correlation Coefficient ◽  
Geographical Location ◽  
Small River ◽  
Binary Relations ◽  
Catchment Basin ◽  
Space Images ◽  
The North ◽  
Inverse Effect ◽  
Characteristic Points ◽  
Fractal Representation

The purpose of the article is the analysis of asymmetric wavelets in binary relations between three coordinates at 290 characteristic points from the source to the mouth of the small river Irovka. The hypsometric characteristic is the most important property of the relief. The Irovka River belongs to a low level, at the mouth it is 89 m high, and at the source it is 148 m above sea level. Modeling of binary relations with latitude, longitude, and height has shown that local latitude receives the greatest quantum certainty. In this case, all paired regularities received a correlation coefficient of more than 0.95. Such a high adequacy of wave patterns shows that geomorphology can go over to the wave multiple fractal representation of the relief. The Irovka River is characterized by a small anthropogenic impact, therefore, the relief over a length of 69 km has the natural character of the oscillatory adaptation of a small river to the surface of the Vyatka Uval from its eastern side. This allows us to proceed to the analysis of the four tributaries of the small river Irovka, as well as to model the relief of the entire catchment basin of 917 km2. The greatest adequacy with a correlation coefficient of 0.9976 was obtained by the influence of latitude on longitude, that is, the geographical location of the relief of the river channel with respect to the geomorphology of the Vyatka Uval. In second place with a correlation of 0.9967 was the influence of the height of the points of the channel of the small river on local longitude and it is also mainly determined by the relief of the Vyatka Uval. In third place was the effect of latitude on height with a correlation coefficient of 0.9859. And in last sixth place is the inverse effect of altitude on local latitude in the North-South direction.

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.

RELEVANCE LOGIC AND THE CALCULUS OF RELATIONS

2010 ◽  
Vol 3 (1) ◽  
pp. 41-70 ◽  
Author(s):  
ROGER D. MADDUX
Keyword(s):  
Propositional Logic ◽  
Relevance Logic ◽  
Binary Relations ◽  
Classical Propositional Logic

Sound and complete semantics for classical propositional logic can be obtained by interpreting sentences as sets. Replacing sets with commuting dense binary relations produces an interpretation that turns out to be sound but not complete for R. Adding transitivity yields sound and complete semantics for RM, because all normal Sugihara matrices are representable as algebras of binary relations.

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