multiplex (version 3.7)

cngr: Congruence Relations

Description

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

Usage

cngr(S, PO = NULL, uniq)

Value

An object of ‘Congruence’ class. The items included are:

S

semigroup of relations

PO

partial order table (if specified)

clu

congruence classes

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

Author

Antonio Rivero Ostoic

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.

References

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

See Also

decomp, fact, pacnet

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 ) )

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

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