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### Distributive laws for commuting equivalence relations

Catherine Huafei Yan
1998 Discrete Mathematics
We give some necessary and sufficient conditions 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  ...

### The Theory of Commuting Subalgebras of Complete Heyting Algebras

Jeffrey Thomas Crants, Catherine Huafei Yan
that the cut elimination axiom is essentially 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.  ...

### Corelations are the prop for extraspecial commutative Frobenius monoids [article]

Brandon Coya, Brendan Fong
2016 arXiv   pre-print
Dually, we show that the category of 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  ...

### Relations among Matrices over a Semiring [chapter]

Dylan Killingbeck, Milene Santos Teixeira, Michael Winter
2015 Lecture Notes in Computer Science
(Annihilator 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?  ...

### 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  ...

### Congruence Modularity Implies the Arguesian Law for Single Algebras with a Difference Term

Paolo Lipparini
1999 Journal of Algebra
Recently, a generalization of 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.  ...

### On commutative weak BCK-algebras [article]

Janis Cirulis
2015 arXiv   pre-print
In particular, 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  ...

### The Algebra of Partial Equivalence Relations

Fabio Zanasi
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 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.  ...

### Modern Set [article]

Jun Tanaka
2008 arXiv   pre-print
., the 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 ).  ...

### 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.  ...

### Book Review: Commutator theory for congruence modular varieties

Stanley Burris
1989 Bulletin of the American Mathematical Society
in the variety has a lattice of congruences satisfying the 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.  ...

### The Theory of Commuting Boolean Sigma-Algebras

Catherine Huafei Yan
lattices of 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  ...

2017 arXiv   pre-print
Finally, a similar theorem 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.  ...

### Universal Associative Geometry [article]

Wolfgang Bertram
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 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  ...

### PROPs for involutive monoids and involutive bimonoids [article]

Daniel Graves
2021 arXiv   pre-print
We prove that the category of involutive bimonoids in a symmetric monoidal category is 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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