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.
12
questions
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 ...
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 ...
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\...
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 ...
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 ...
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 ...
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 ...
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
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 ...
1
vote
0
answers
18
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 ...
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 ...
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 $subst(r,s)$ ...