cngr: Congruence Relations

In multiplex: Algebraic Tools for the Analysis of Multiple Social Networks

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 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)

multiplex documentation built on Sept. 30, 2024, 5:07 p.m.