multiplex (version 4.0)

cngr: Congruence Relations

Description

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

Usage

cngr(S, PO, uniq)

Value

A “Congruence” class list object with:

S

Semigroup of relations

PO

Partial order table (if specified)

clu

Congruence classes

Arguments

S

A “Semigroup” class object.

PO

(optional) A partial order table.

uniq

(optional and logical) Return unique congruence relations?

Details

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

In case that the partial order is supplied in PO the computation of the congruence classes is slightly faster than for an abstract semigroup.

References

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

See Also

decomp, fact, pacnet

Examples

# create 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 ) )

# abstract semigroup of 'arr'
S <- arr |> semigroup()

# look at the congruences in S
cngr(S)