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Overlap Algebras: a Constructive Look at Complete Boolean Algebras [article]

Francesco Ciraulo, Michele Contente
2020 arXiv   pre-print
The notion of a complete Boolean algebra, although completely legitimate in constructive mathematics, fails to capture some natural structures such as the lattice of subsets of a given set.  ...  Sambin's notion of an overlap algebra, although classically equivalent to that of a complete Boolean algebra, has powersets and other natural structures as instances.  ...  The latter fails constructively; a constructive version can be obtained by replacing complete Boolean algebras by Sambin's overlap algebras, as we now see. Definition 2.1.  ... 
arXiv:1904.13320v3 fatcat:sd5z454gqbbyrpm3bxu27xk5pi

Overlap Algebras: a Constructive Look at Complete Boolean Algebras

Francesco Ciraulo, Michele Contente
2019 Logical Methods in Computer Science  
The notion of a complete Boolean algebra, although completely legitimate in constructive mathematics, fails to capture some natural structures such as the lattice of subsets of a given set.  ...  Sambin's notion of an overlap algebra, although classically equivalent to that of a complete Boolean algebra, has powersets and other natural structures as instances.  ...  The latter fails constructively; a constructive version can be obtained by replacing complete Boolean algebras by Sambin's overlap algebras, as we now see. Definition 2.1.  ... 
doi:10.23638/lmcs-16(1:13)2020 fatcat:lr2val2apfgjvcmell2eoyljq4

Overlap Algebras as Almost Discrete Locales [article]

Francesco Ciraulo
2021 arXiv   pre-print
Our claim is that Sambin's Overlap Algebras have good enough features to deserve to be called that.  ...  Boolean locales are "almost discrete", in the sense that a spatial Boolean locale is the same thing as the frame of open subsets of a discrete space, that is, the powerset of a set.  ...  Introduction If the open sets of a topological space form a complete Boolean algebra, instead of a mere complete Heyting algebra, then the space is "almost discrete": every Kolmogorov almost discrete space  ... 
arXiv:1601.04830v2 fatcat:5adeucvv6rgiho6vwt47ijrfeq

The overlap algebra of regular opens

Francesco Ciraulo, Giovanni Sambin
2010 Journal of Pure and Applied Algebra  
Since they are a constructive counterpart to complete Boolean algebras and, at the same time, they have a more powerful axiomatization than Heyting algebras, overlap algebras are expected to turn out useful  ...  Overlap algebras can be seen as particular open (or overt) locales and, from a classical point of view, they essentially coincide with complete Boolean algebras.  ...  In particular, it was he who first noticed the connection with open locales and with complete Boolean algebras.  ... 
doi:10.1016/j.jpaa.2010.02.002 fatcat:lurhv2gx5re5han2n3mnzfqbqm

Formalising Overlap Algebras in Matita

CLAUDIO SACERDOTI COEN, ENRICO TASSI
2011 Mathematical Structures in Computer Science  
Overlap Algebras are new algebraic structures designed to ease reasoning about subsets in an algebraic way within intuitionistic logic.  ...  We describe some formal topological results, formalized in Matita 1/2, presented in predicative intuitionistic logic and in terms of Overlap Algebras.  ...  Overlap Algebras are to intuitionistic subset theory what Boolean algebras are to classical subset theory.  ... 
doi:10.1017/s0960129511000107 fatcat:7mqwg6ryrvfj5dvswlp7wnmk7q

Intuitionistic overlap structures

Francesco Ciraulo
2013 Logic and Logical Philosophy  
The main thesis of this paper is that the overlap relation in the latter sense is a necessary tool for a constructive and intuitionistic development of point-free geometry.  ...  We study some connections between two kinds of overlap relations: that of point-free geometries in the sense of Grzegorczyk, Whitehead and Clarke, and that recently introduced by Sambin within his constructive  ...  It is a fact of life that Pow(S) is no longer a Boolean algebra as long as you look at it from an intuitionistic point of view.  ... 
doi:10.12775/llp.2013.011 fatcat:hhir5heupvc2xhcf4tzthkqaxu

Regular opens in constructive topology and a representation theorem for overlap algebras

Francesco Ciraulo
2013 Annals of Pure and Applied Logic  
Giovanni Sambin has recently introduced the notion of an overlap algebra in order to give a constructive counterpart to a complete Boolean algebra.  ...  In particular we show that there exists a duality between the category of set-based overlap algebras and a particular category of topologies in which all open subsets are regular.  ...  With classical logic, overlap algebras are precisely complete Boolean algebras and hence the notion of an overlap algebra is strictly stronger than that of a complete Heyting algebra.  ... 
doi:10.1016/j.apal.2012.10.006 fatcat:gmp4gog6ivci3f3kyhmcxjjkf4

Small overlap monoids II: Automatic structures and normal forms

Mark Kambites
2009 Journal of Algebra  
We show that any finite monoid or semigroup presentation satisfying the small overlap condition C (4) has word problem which is a deterministic rational relation.  ...  It follows that the set of lexicographically minimal words forms a regular language of normal forms, and that these normal forms can be computed in linear time.  ...  Remeslennikov for a number of helpful conversations.  ... 
doi:10.1016/j.jalgebra.2008.12.028 fatcat:u7pqoqnjgvgd7atr4h62v2qj3m

A Cube Algebra with Comparative Operations: Containment, Overlap, Distance and Usability [article]

Panos Vassiliadis
2022 arXiv   pre-print
new cube from a previous one, defined at a different level of abstraction.  ...  In this paper, we provide a comprehensive rigorous modeling for multidimensional spaces with hierarchically structured dimensions in several layers of abstractions and data cubes that live in such spaces  ...  We thus need a mechanism to detect inclusion, overlap and non-overlap of queries at the detailed level, at least as a pre-requisite to facilitate the assessment of novelty.  ... 
arXiv:2203.09390v1 fatcat:qvqjwxc7sjfcve4d32nkd432ty

Partiality II: Constructed relation algebras

Gunther Schmidt
2012 The Journal of Logic and Algebraic Programming  
It will be shown how these can be fruitfully applied concerning constructions with temporarily non-connected relation algebras.  ...  Matrices of partialities will considerably improve the possibility to study non-strictness, streams, partial evaluation, and net properties in a compact relation-algebraic form.  ...  First, we take a look at a very primitive construction. Proposition.  ... 
doi:10.1016/j.jlap.2012.05.005 fatcat:kxx3hpgzwjekpe2bvifvxk4bxq

Axioms, Algebras and Topology [chapter]

Brandon Bennett, Ivo Düntsch
2007 Handbook of Spatial Logics  
Another way of looking at these algebras is via a certain class of lattices: An algebra A, +, ·, ⇒, 0, 1 of type 2, 2, 2, 0, 0 is called a Heyting algebra (or pseudo-Boolean algebra [89] ) if A, +, ·  ...  We have looked at the relationships between algebraic models, point-set topology and axiomatic theories of spatial regions.  ... 
doi:10.1007/978-1-4020-5587-4_3 fatcat:ano2fojxs5c3lo2sbdrkqxwsg4

Strongly representable algebras [article]

Tarek Sayed Ahmed
2013 arXiv   pre-print
We give a simpler proof of a result of Hodkinson in the context of a blow and blur up construction argueing that the idea at heart is similar to that adopted by Andréka et all sayed.  ...  This construction, which works for both relation and cylindric algebras, further shows that the class of strongly representable atom structures is not elementary.  ...  As before, for each u ∈ At, choose any countable Boolean elementary complete subalgebra of A u , B u say. Le u i : i < m be elements in At, and let v = τ (u 1 , . . . u m ).  ... 
arXiv:1305.4532v1 fatcat:7oac6zr3wjbtvihfyaocqurxr4

Orthomodular algebraic lattices related to combinatorial posets

Luca Bernardinello, Lucia Pomello, Stefania Rombolà
2014 Italian Conference on Theoretical Computer Science  
An orthomodular lattice is always regular ( [7] ) and hence can be seen as a family of partially overlapping Boolean algebras.  ...  Orthomodular lattices are families of partially overlapping Boolean algebras and have been studied as the algebraic model of quantum logic [7] .  ...  Hence, if A ⊂ B then B contains at least an element x concurrent with A. The orthomodular law is weaker than the distributive law. Orthocomplemented distributive lattices are called Boolean algebras.  ... 
dblp:conf/ictcs/BernardinelloPR14 fatcat:3wuka6uf55fctpfdj6p2v2os54

Algebraic Neighbourhood Logic

Peter Höfner, Bernhard Möller
2008 The Journal of Logic and Algebraic Programming  
A further simplification is achieved, since the semiring methods used are easy and fairly standard.  ...  We present an algebraic embedding of Neighbourhood Logic (NL) into the framework of semirings which yields various simplifications.  ...  More precisely we have a look at the chop operator and at Allen's 13 relations between intervals.  ... 
doi:10.1016/j.jlap.2007.10.004 fatcat:svugk277zvfxnpfzuwoygqwemi

Chain homogeneous Souslin algebras [article]

Gido Scharfenberger-Fabian
2010 arXiv   pre-print
Assuming Jensen's principle diamond-plus we construct Souslin algebras all of whose maximal chains are pairwise isomorphic as total orders, thereby answering questions of Koppelberg and Todorcevic  ...  Thanks are due to Sabine Koppelberg for asking the initial question, to Stevo Todorçević for asking a question which led to a significant improvement of my results and to Peter Krautzberger and Stefan  ...  The main reference for looking up undefined notions is the first volume of the Handbook of Boolean Algebras, [Kop89] . Let B be a Boolean algebra.  ... 
arXiv:1005.4948v1 fatcat:24cbla4pt5bwbj2lcjjdybntdi
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