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.

Filter by
Sorted by
Tagged with
1 vote
0 answers
19 views

Aggregation relationship in class diagram

How do you transform an aggregation relatinship between to classes in a class diagram (1 to many or many to many) into relational schema and is it needed/possible. Or is enough if we transform only ...
2 votes
0 answers
248 views

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 ...
4 votes
0 answers
135 views

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
196 views

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\...
1 vote
1 answer
70 views

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 ...
10 votes
1 answer
661 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 ...
73 votes
13 answers
6k views

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 ...
2 votes
2 answers
640 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 ...
1 vote
0 answers
82 views

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
1 answer
384 views

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
2 answers
588 views

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 ...
1 vote
0 answers
170 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 $subst(r,s)$ ...