green.rel {multiplex}

Green's Relations of Abstract Semigroups

Description

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

Usage

green.rel(S)

Arguments

S

an object of a 'Semigroup' class

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.

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

Note

Some systems have the D-class equal to S.

Author(s)

Antonio Rivero Ostoic

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: 2 binary relations among 3 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()