multiplex (version 3.7)

green.rel: Green's Relations of Abstract Semigroups

Description

A function to produce the Green's relations of a semigroup object.

Usage

green.rel(S)

Value

A list with the abstract semigroup, clustering of equivalence classes, and egg-box diagram that are separated by “|” and “-”.

S

multiplication matrix of the input semigroup

ord

dimension of the semigroup

st

vector of the unique string relations

clu

list of vectors with clustering information for \(R\) and \(L\) classes

R

\(R\) equivalence classes

L

\(L\) equivalence classes

D

\(D\) equivalence classes

Arguments

S

an object of a ‘Semigroup’ class

Author

Antonio Rivero Ostoic

Details

Function green.rel produces the egg-box diagram (Green, 1951) of an abstract semigroup \(S\), which is the union of the left compatible \(R\) equivalence and the right compatible \(L\) equivalence classes that makes the \(D\)-class on \(S\).

References

Green, J. “On the structure of semigroups,” Annals of Mathematics 54(1), 163--172, 1951.

Ostoic, JAR “Relational systems of transport network and provinces in ancient Rome,” in Mathematics for social sciences and arts -- algebraic modeling. Springer Nature. 2023.

See Also

semigroup, diagram, as.semigroup, edgeT, wordT, fact, cngr, decomp

Examples

# create the data: two binary relations among three elements
arr <- round( replace( array(runif(18), c(3,3,2)), array(runif(18),
       c(3,3,2))>.5, 1 ) )
# optional string labels
dimnames(arr)[[3]] <- list("n", "m")

# look at the semigroup and its Green's relations
semigroup(arr) |> 
  green.rel()