VARIETIES OF EQUALITY STRUCTURES

2003 ◽  
Vol 13 (04) ◽  
pp. 463-480 ◽  
Author(s):  
DESMOND FEARNLEY-SANDER ◽  
TIM STOKES
Keyword(s):  
Binary Operation ◽  
Topological Spaces ◽  
Natural Conditions ◽  
Universal Algebras ◽  
Natural Way

We consider universal algebras which are monoids and which have a binary operation we call internalized equality, satisfying some natural conditions. We show that the class of such E-structures has a characterization in terms of a distinguished submonoid which is a semilattice. Some important varieties (and variety-like classes) of E-structures are considered, including E-semilattices (which we represent in terms of topological spaces), E-rings (which we show are equivalent to rings with a generalized interior operation), E-quantales (where internalized equalities on a fixed quantale in which 1 is the largest element are shown to correspond to sublocales of the quantale), and EI-structures (in which an internalized inequality is defined and interacts in a natural way with the equality operation).

scholarly journals Topological characterizations of amenability and congeniality of bases

2020 ◽  
Vol 21 (1) ◽  
pp. 1
Author(s):  
Sergio R. López-Permouth ◽  
Benjamin Stanley
Keyword(s):  
Direct Product ◽  
Topological Spaces ◽  
Countable Dimension ◽  
One To One ◽  
Natural Way

<div>We provide topological interpretations of the recently introduced notions of amenability and congeniality of bases of innite dimensional algebras. In order not to restrict our attention only to the countable dimension case, the uniformity of the topologies involved is analyzed and therefore the pertinent ideas about uniform topological spaces are surveyed.</div><div><p>A basis B over an innite dimensional F-algebra A is called amenable if F<sup>B</sup>, the direct product indexed by B of copies of the eld F, can be made into an A-module in a natural way. (Mutual) congeniality is a relation that serves to identify cases when different amenable bases yield isomorphic A-modules.</p><p>(Not necessarily mutual) congeniality between amenable bases yields an epimorphism of the modules they induce. We prove that this epimorphism is one-to-one only if the congeniality is mutual, thus establishing a precise distinction between the two notions.</p></div>

scholarly journals Residuation in non-associative MV-algebras

2018 ◽  
Vol 68 (6) ◽  
pp. 1313-1320
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Binary Operation ◽  
Residuated Lattice ◽  
Natural Question ◽  
Natural Conditions ◽  
Double Negation ◽  
Mv Algebra ◽  
Residuated Poset ◽  
Double Negation Law ◽  
Mv Algebras

Abstract It is well known that every MV-algebra can be converted into a residuated lattice satisfying divisibility and the double negation law. In a previous paper the first author and J. Kühr introduced the concept of an NMV-algebra which is a non-associative modification of an MV-algebra. The natural question arises if an NMV-algebra can be converted into a residuated structure, too. Contrary to MV-algebras, NMV-algebras are not based on lattices but only on directed posets and the binary operation need not be associative and hence we cannot expect to obtain a residuated lattice but only an essentially weaker structure called a conditionally residuated poset. Considering several additional natural conditions we show that every NMV-algebra can be converted in such a structure. Also conversely, every such structure can be organized into an NMV-algebra. Further, we study an a bit more stronger version of an algebra where the binary operation is even monotone. We show that such an algebra can be organized into a residuated poset and, conversely, every residuated poset can be converted in this structure.

scholarly journals Quasi-uniform completions of partially ordered spaces

2007 ◽  
Vol 57 (2) ◽  
Author(s):  
David Buhagiar ◽  
Tanja Telenta
Keyword(s):  
Topological Space ◽  
Partial Order ◽  
Uniform Space ◽  
Uniform Spaces ◽  
Natural Conditions ◽  
Ordered Topological Space ◽  
Partially Ordered ◽  
Partially Ordered Topological Space ◽  
Partially Ordered Spaces ◽  
Natural Way

AbstractIn this paper we define partially ordered quasi-uniform spaces (X, $$\mathfrak{U}$$ , ≤) (PO-quasi-uniform spaces) as those space with a biconvex quasi-uniformity $$\mathfrak{U}$$ on the poset (X, ≤) and give a construction of a (transitive) biconvex compatible quasi-uniformity on a partially ordered topological space when its topology satisfies certain natural conditions. We also show that under certain conditions on the topology $$\tau _{\mathfrak{U}*} $$ of a PO-quasi-uniform space (X, $$\mathfrak{U}$$ , ≤), the bicompletion $$(\tilde X,\tilde {\mathfrak{U}})$$ of (X, $$\mathfrak{U}$$ ) is also a PO-quasi-uniform space ( $$(\tilde X,\tilde {\mathfrak{U}})$$ , ⪯) with a partial order ⪯ on $$\tilde X$$ that extends ≤ in a natural way.

scholarly journals Agrarian structure changes in the village farmsteads on Durmitor

2021 ◽  
Vol 68 (1) ◽  
pp. 11-21
Author(s):  
Darko Stijepović
Keyword(s):  
Agricultural Land ◽  
Arable Land ◽  
Agricultural Land Use ◽  
Natural Conditions ◽  
Structure Changes ◽  
Basic Production ◽  
Land Surfaces ◽  
The Village ◽  
Natural Way ◽  
Agrarian Structure

The paper presents findings about agrarian structure changes that have occurred in the village farmsteads on Durmitor highlands considering thesis that this area has experienced complex and strong changes. The most important characteristics of these changes relate with , aging, de-agrarian striving; and reduction of the livestock fund, land surfaces and number of homesteads. The agrarian structure of the Durmitor area, which includes the municipalities of Žabljak, Šavnik and Plužine, is characterized by the dominant participation of individual farms focused on sheep and catlle production.The structure of agricultural land use in the examined region was defined by natural conditions and natural way of production. In the total agricultural area, natural meadows and pastures occupy over 90% of the area, and arable land and orchards occupy a small percentage of the area.The existing land structure determined livestock production as the basic production.

Neighborhoods and convergence with respect to a closure operator

2011 ◽  
Vol 61 (5) ◽  
Author(s):  
Josef Šlapal
Keyword(s):  
Closure Operator ◽  
Topological Spaces ◽  
Filter Convergence ◽  
Categorical Closure Operator ◽  
Natural Way

AbstractWe study neighborhoods with respect to a categorical closure operator. In particular, we discuss separation and compactness obtained from neighborhoods in a natural way and compare them with the usual closure separation and closure compactness. We also introduce a concept of convergence based on using centered systems of subobjects, which naturally generalizes the classical filter convergence in topological spaces. We investigate behavior of the convergence introduced and show, among others, that it relates to the separation and compactness in natural ways.

On endomorphisms of power-semigroups

2017 ◽  
Vol 10 (03) ◽  
pp. 1750058 ◽  
Author(s):  
Yeni Susanti ◽  
Joerg Koppitz
Keyword(s):  
Binary Operation ◽  
Identity Element ◽  
Orthodox Semigroup ◽  
Natural Way

An involuted semilattice [Formula: see text] is a semilattice [Formula: see text] with an identity element [Formula: see text] and with an involution [Formula: see text] satisfying [Formula: see text] and [Formula: see text]. We consider involuted semilattices [Formula: see text] with an identity [Formula: see text] such that there is a subsemilattice [Formula: see text] without [Formula: see text] with the property that any [Formula: see text] belongs to exactly one of the following four sets : [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text]. In this paper, we introduce an associative binary operation [Formula: see text] on [Formula: see text] in the following quite natural way: [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] for [Formula: see text] and characterize all endomorphisms of the orthodox semigroup [Formula: see text].

scholarly journals Derived length for arbitrary topological spaces

1992 ◽  
Vol 15 (2) ◽  
pp. 273-277 ◽  
Author(s):  
A. J. Jayanthan
Keyword(s):  
Topological Spaces ◽  
Derived Length ◽  
Ordinal Numbers ◽  
Scattered Spaces ◽  
Natural Way

The notion of derived length is as old as that of ordinal numbers itself. It is also known as the Cantor-Bendixon length. It is defined only for dispersed (that is scattered) spaces. In this paper this notion has been extended in a natural way for all topological spaces such that all its pleasing properties are retained. In this process we solve a problem posed by V. Kannan. ([1] Page 158).

scholarly journals Green's Relations on HypG(2)

2012 ◽  
Vol 20 (1) ◽  
pp. 249-264
Author(s):  
Wattapong Puninagool ◽  
Sorasak Leeratanavalee
Keyword(s):  
Binary Operation ◽  
Green’S Relations ◽  
Operation Symbol ◽  
Green's Relations ◽  
Natural Way

AbstractA generalized hypersubstitution of type τ = (2) is a mapping which maps the binary operation symbol f to a term σ(f) which does not necessarily preserve the arity. Any such σ can be inductively extended to a map σ̂ on the set of all terms of type τ = (2), and any two such extensions can be composed in a natural way. Thus, the set HypG(2) of all generalized hypersubstitutions of type τ = (2) forms a monoid. Green's relations on the monoid of all hypersubstitutions of type τ = (2) were studied by K. Denecke and Sh.L. Wismath. In this paper we describe the classes of generalized hypersubstitutions of type τ = (2) under Green's relations.

scholarly journals A Topological Approach to Chemical Organizations

2009 ◽  
Vol 15 (1) ◽  
pp. 71-88 ◽  
Author(s):  
Gil Benkö ◽  
Florian Centler ◽  
Peter Dittrich ◽  
Christoph Flamm ◽  
Bärbel M. R. Stadler ◽  
...  
Keyword(s):  
Chemical Species ◽  
Alternative Interpretation ◽  
Point Of View ◽  
Topological Spaces ◽  
Production Rules ◽  
Topological Approach ◽  
Distinctive Features ◽  
The Individual ◽  
Directed Hypergraph ◽  
Natural Way

Large chemical reaction networks often exhibit distinctive features that can be interpreted as higher-level structures. Prime examples are metabolic pathways in a biochemical context. We review mathematical approaches that exploit the stoichiometric structure, which can be seen as a particular directed hypergraph, to derive an algebraic picture of chemical organizations. We then give an alternative interpretation in terms of set-valued set functions that encapsulate the production rules of the individual reactions. From the mathematical point of view, these functions define generalized topological spaces on the set of chemical species. We show that organization-theoretic concepts also appear in a natural way in the topological language. This abstract representation in turn suggests the exploration of the chemical meaning of well-established topological concepts. As an example, we consider connectedness in some detail.

I-Subsemigroups and α-monomorphisms

1973 ◽  
Vol 16 (2) ◽  
pp. 146-166 ◽  
Author(s):  
Kenneth D. Magill
Keyword(s):  
Topological Space ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Natural Way

To each idempotent v of a semigroup T, there is associated, in a natural way, a subsemigroup Tv of T. The subsemigroup Tv is simply the collection of all elements of T for which v acts as a two-sided identity. We refer to such a subsemigroup as an I-subsemigroup of T. We first establish some elementary properties of these subsemigroups with no restrictions on the semigroup in which they are contained. Then we turn our attention to the semigroup of all continuous selfmaps of a topological space. The I-subsemigroups of these semigroups are investigated in some detail and so are the a-monomorphisms [3, p. 518] from one such semigroup into another. Among other things, a relationship is established between I-subsemigroups and α-monomorphisms. An analogous theory exists for semigroups of closed selfmaps on topological spaces. A number of results are listed for these semigroups with the proofs often deleted since, in many cases, the situation is much the same as for semigroups of continuous functions.

Sign in / Sign up

Export Citation Format

Share Document