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Distributive laws for commuting equivalence relations

1998
*
Discrete Mathematics
*

We give some necessary and sufficient conditions

doi:10.1016/s0012-365x(97)00061-7
fatcat:sczshfdpfjf3firn5c3l6t3lbq
*for*the*distributive**law*and its dual to hold*for**commuting**equivalence**relations*. (~) 1998 Elsevier Science B.V. ... In the present paper, we consider*commuting**equivalence**relations*. It has been proved by J6nsson (1953) that the modular*law*holds in the lattice of*commuting**equivalence**relations*. ... Acknowledgements 1 am greatly indebted to my advisor Professor Gian-Carlo Rota*for*teaching me the theory of*equivalence**relations*, and*for*his endless supporting and encouragement in doing this problem ...##
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The Theory of Commuting Subalgebras of Complete Heyting Algebras

1998
*
Advances in Mathematics
*

that the cut elimination axiom is essentially

doi:10.1006/aima.1998.1758
fatcat:gx4sopduqzevfmkyoaijfcxhti
*equivalent*to the*distributive**law*. ... It is the natural generalization of the notion of*commutativity**for**equivalence**relations*. Definition 3.1. ... To obtain a completeness theorem*for*Horn sentences with a countable set of implications, utilize the proof of Theorem 6.9. ...##
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Corelations are the prop for extraspecial commutative Frobenius monoids
[article]

2016
*
arXiv
*
pre-print

Dually, we show that the category of

arXiv:1601.02307v2
fatcat:ox3epfa5hrdf5lzdgcqxqchdr4
*relations*is*equivalent*to the prop*for*special*commutative*bimonoids. Throughout, we emphasise how corelations model interconnection. ... We define the category of corelations between finite sets and prove that it is*equivalent*to the prop*for*extraspecial*commutative*Frobenius monoids. ... Corelations are the prop*for*extraspecial*commutative*Frobenius monoids In the influential paper [La04] , Lack develops the theory of*distributive**laws**for*props, and proves the following as an example ...##
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Relations among Matrices over a Semiring
[chapter]

2015
*
Lecture Notes in Computer Science
*

(Annihilator

doi:10.1007/978-3-319-24704-5_7
fatcat:j73gw5g2ffdfrhlodcdadkswku
*Law*) A semiring is called*commutative*if * is*commutative*, i.e., if we have x * y = y * x*for*all x, y ∈ S. ... Let ≡ be the*equivalence**relation*that has the three*equivalence*classes [0] = {0}, [1] = {1} and [n] = {n ∈ N | n > 1}. ... 1 The following two conditions are*equivalent*: Thank you*for*your attention! Questions? ...##
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Page 306 of Annals of Mathematics Vol. 43, Issue 2
[page]

1942
*
Annals of Mathematics
*

In fact, (i) is not really essential, if we are willing to introduce an

*equivalence**relation*. 9. ... 306 GARRETT BIRKHOFF*equivalence**relation*, which is a congruence*relation*with respect to the addition defined by (16), and relative to which the latter is associative, and gives any element b — c an inverse ...##
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Congruence Modularity Implies the Arguesian Law for Single Algebras with a Difference Term

1999
*
Journal of Algebra
*

Recently, a generalization of

doi:10.1006/jabr.1999.7874
fatcat:pp6evtct3jeohabw3nmhtva6jm
*commutator*theory has been developed*for*algebraic systems belonging to a congruence modular variety. ... This general*commutator*theory is used here both to provide a very simple proof of a classical result by R. Freese and B. Jonsson and to solve a problem raised by S. Tschantz. ... ACKNOWLEDGMENT We are grateful to the referee*for*many interesting and useful comments and suggestions, and in particular*for*communicating to us the result in Proposition 5.3. ...##
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On commutative weak BCK-algebras
[article]

2015
*
arXiv
*
pre-print

In particular,

arXiv:1304.0999v2
fatcat:xmzlf2c2hvdxbcvq7aoesejxpa
*commutative*weak BCK-algebras are just those meet semilattices with the least element in which all initial segments are non-*distributive*de Morgan lattices. ... in connection with various quantum logics are, in fact,*commutative*weak BCK-algebras belonging to that or other of these classes. ... The following assertions are*equivalent*in any*commutative*wBCKalgebra: Lemma 4 . 1 . 41 The Pierce*law*(− 18 ), the contraction*law*(− 15 ) and the contraction rule (− 16 ) are*equivalent*in*commutative*...##
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The Algebra of Partial Equivalence Relations

2016
*
Electronical Notes in Theoretical Computer Science
*

This paper shows how the same construction can be used in a cartesian setting to obtain presentations by generators and equations

doi:10.1016/j.entcs.2016.09.046
fatcat:ifb4n2rau5eolhfrjly4qox2a4
*for*the PROP of*equivalence**relations*and of partial*equivalence**relations*... structures interact according to PROP operations of sum, fibered sum and composition via a*distributive**law*. ... A presentation of*equivalence**relations*This section builds modularly a presentation*for*the PROP ER of*equivalence**relations*, using the operations introduced in § 3. ...##
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Modern Set
[article]

2008
*
arXiv
*
pre-print

., the

arXiv:0806.2943v1
fatcat:onqjvk5ndbhdngdalzb5ztw3ru
*law*of excluded middle, the*law*of non-contradiction, the*distributive**law*, the*commutative**law*,etc.... ... ., the*law*of excluded middle, the*law*of non-contradiction, the*distributive**law*, the*commutative**law*,etc.... ... We can show the*commutative**law**for*intersection similarly. We will show the two most important cases of the*distributive**law*briefly. Call C = A ∨ ( B ∧ C ). ...##
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Page 182 of Computational Linguistics Vol. 19, Issue 1
[page]

1993
*
Computational Linguistics
*

Similar

*equivalences*based on logical properties of other operators—including associativity and*commutativity*of disjunction, de Morgan’s*laws*,*laws*of*distributivity*, equiva- lence of quantifier prefixes ... Intuitively at least, more logical forms should be*equivalent*than are provably*equivalent*by the*laws*of the logic alone. ...##
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Book Review: Commutator theory for congruence modular varieties

1989
*
Bulletin of the American Mathematical Society
*

in the variety has a lattice of congruences satisfying the

doi:10.1090/s0273-0979-1989-15710-8
fatcat:klzp53752faqti3rwrzmdhy6by
*distributive**law*. ... A congruence of an algebra A is an*equivalence**relation*such that the obvious definition of a quotient algebra works; congruences can also be thought of as kernels of homomorphisms. ... in the variety has a lattice of congruences satisfying the*distributive**law*. ...##
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The Theory of Commuting Boolean Sigma-Algebras

1999
*
Advances in Mathematics
*

lattices of

doi:10.1006/aima.1998.1823
fatcat:ehz37t735reifctoe6jxe5dria
*commuting**equivalence**relations*. ... Two*equivalence**relations*R and T*commute*if and only of R b T is an*equivalence**relation*. ... are qualitatively independent; if`E 1 ,`E 2*commute*as*equivalence**relations*of M, then f B, C (`E 1 ), f B, C (`E 2 )*commute*as*equivalence**relations*of M$, and f B, The space M$ is called the completion ...##
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Monad and Comonad Objects through 2-adjunctions of the type Adj-Mnd
[article]

2017
*
arXiv
*
pre-print

Finally, a similar theorem

arXiv:1510.04724v3
fatcat:hot6xr4gtfek5jwb7txhm2a7cm
*relating*mixed*distributive**laws*with a pair of Eilenberg-Moore liftings on algebras and coalgebras is proved, with a naturality*for*the*equivalence*on the monad and comonad ... In this article, the author analyses*distributive*and mixed*distributive**laws*and some of their*equivalences*through the use of 2-adjunctions of the type -. ... Acknowledgement The author would like to thank to the Consejo Nacional de Ciencia y Tecnología (CONACYT)*for*the financial support through the grant SNI-59154. ...##
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Universal Associative Geometry
[article]

2014
*
arXiv
*
pre-print

We reinvestigate the case of homogeneous pregroupoids (corresponding to the projective geometry of a group) from the point of view of pairs of

arXiv:1406.1692v1
fatcat:zhtpcwz7obeqrezrlzx4ubl5py
*commuting*principal*equivalence**relations*. ...*relations*, have a natural prolongation from a set to its the power set. ... We will show that they are*related*by a certain*distributive**law*, by using the following description of the product (xyz) ab in the "affine picture": Let (a, b) be two*commuting*prev's on Ω, and fix ...##
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PROPs for involutive monoids and involutive bimonoids
[article]

2021
*
arXiv
*
pre-print

We prove that the category of involutive bimonoids in a symmetric monoidal category is

arXiv:2005.07968v2
fatcat:ufkpugmen5bs7e2dvwxcxeopsu
*equivalent*to the category of algebras over a PROP constructed from the category of involutive non-*commutative*sets ... The category of involutive non-*commutative*sets encodes the structure of an involution compatible with a (co)associative (co)multiplication. ... Acknowledgements I would like to thank Sarah Whitehouse*for*all her guidance and encouragement. ...
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