cngr {multiplex}

Congruence Relations

Description

Find the congruence relations of a given abstract or a partially ordered semigroup.

Usage

cngr(S, PO = NULL, uniq)

Arguments

S

an object from the 'Semigroup' class.

PO

(optional) the partial order table

uniq

(optional and logical) whether or not return the unique congruence relations

Details

Congruencies are equivalence relations that preserve the operation between the correspondent classes in the algebraic structure. In this case, the different congruence classes are based on the substitution property of the semigroup object.

Value

An object of 'Congruence' class. The items included are:

S

semigroup of relations

PO

partial order table (if specified)

clu

congruence classes

Note

If the partial order is supplied in the input, then the computation of the congruence classes is slightly faster than for an abstract semigroup.

Author(s)

Antonio Rivero Ostoic

References

Hartmanis, J. and R.E. Stearns Algebraic Structure Theory of Sequential Machines. Prentice-Hall. 1966.

See Also

decomp, fact, pacnet

Examples

## Create an abstract semigroup object
arr <- round( replace( array(runif(18), c(3,3,2)), array(runif(18),
       c(3,3,2))>.5, 1 ) )
#
S <- semigroup(arr)

## look at the congruences in S
cngr(S, PO=NULL)