Questions tagged [relation-algebra]

A relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation. The motivating example of a relation algebra is the set of binary relations on a set X, that is, the set of subsets of X^2.

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Universes from sets of logical relations

Consider any set $I$ and any logical structure $L$. Let $R$ denote some set of $I$-relations over $L$, i.e. each element $r\in R$ sends each $I$-tuple $(x_i)_{i\in I}$ of elements $x_i\in L$ to a ...
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Terminology for the parts of composition?

In function composition, binary relation composition, or more generally category theory, are there distinct names for the two things being composed? If we have $f:X \rightarrow Y$ and $g:Y \rightarrow ...
4 votes
1 answer
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Characterizing relations by forbidden induced subsets

Working with relations in a purely set theoretic manner i.e. as just sets of ordered pairs, we see for any relation $R$ there exists unique inclusion minimal sets $A$ and $B$ such that $R\subseteq A\...
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1 answer
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Generalizing cycle/pseudo-tree factorizations for permutations/transformations to arbitrary binary relations

It's well known every permutation has a unique factorization into disjoint cycles (up to a re-ordering of these factors since they commute), while similarly it can be shown that every transformation ...
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8 votes
1 answer
538 views

Given any finite relation $R$ what is the cardinality of $\langle R\rangle=\{\underbrace{R\circ R\cdots \circ R}_{n\text{ times}}:n\in\mathbb{N}\}$?

Given any finite relation $R$ if we let $\circ$ denote relation composition and define $R^n=\underbrace{R\circ R\cdots \circ R}_{n\text{ times}}$ then does there exist an explicit formula for the ...
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1 vote
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The relation on the set of functions

Let $\varphi: \mathbb{R}^{2} \to \mathbb{R}$ be a symmetric (not necessarily continuous) function (so, $\varphi(x,y)=\varphi(y,x)$ $\forall (x,y)\in \mathbb{R}^{2}$), let $\mathcal{F}$ be the set of ...
2 votes
2 answers
593 views

Categories with binary relations as objects

For the category of functions, pairs of functions making commutative diagrams are the canonical morphisms $\alpha:f\rightarrow g$. For binary relations there is an alternative, to consider the ...
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71 votes
13 answers
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Why is Set, and not Rel, so ubiquitous in mathematics?

The concept of relation in the history of mathematics, either consciously or not, has always been important: think of order relations or equivalence relations. Why was there the necessity of singling ...
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1 answer
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Why is a UNION operation independent in relational algebra?

Why is a set union operation independent in relational algebra? Why it cannot by expressed by the other four basic operations (selection, projection, cartesian product and difference)? What kind of ...
1 vote
0 answers
160 views

Substitution semiring?

Let G be a [ CF ] grammar, and let elements of semiring be sets of rules. Define multiplication as: $$ x\otimes y = \{ t| \exists r \in x \exists s \in y (t=subst(r,s))\} $$ where $subst(r,s)$ ...
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2 answers
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Calculus of Binary Relations

I was reading "Origins of the Calculus of Binary Relations" by Vaughan Pratt where he says "it consists of two components, a logical or static component and a relative or dynamic component" but it ...