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Overlap Algebras: a Constructive Look at Complete Boolean Algebras
[article]
2020
arXiv
pre-print
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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
2019
Logical Methods in Computer Science
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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]
2021
arXiv
pre-print
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2016 pre-print arXiv:1601.04830v1 fatcat:4n3z53qddnc33lv3oikcmazkfa
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
2010
Journal of Pure and Applied Algebra
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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
2011
Mathematical Structures in Computer Science
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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
2013
Logic and Logical Philosophy
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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
2013
Annals of Pure and Applied Logic
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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
2009
Journal of Algebra
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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]
2022
arXiv
pre-print
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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
2012
The Journal of Logic and Algebraic Programming
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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]
2007
Handbook of Spatial Logics
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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]
2013
arXiv
pre-print
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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
2014
Italian Conference on Theoretical Computer Science
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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
2008
The Journal of Logic and Algebraic Programming
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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]
2010
arXiv
pre-print
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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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