decomp: Decomposition of a Semigroup Structure

Description

A function to perform the decomposition of a semigroup structure

Usage

decomp(S, pr, type = c("mca", "pi", "at", "cc"), reduc, fac, force)

Value

An object of ‘Decomp’ class having:

clu: vector with the class membership
eq: the equations in the decomposition
IM: (optional) the image matrices
PO: (optional) the partial order table
ord: (optional) a vector with the order of the image matrices

Arguments

S

an object of a ‘Semigroup’ class

pr

either an object of a ‘Congruence’ class or an object of a ‘Pi.rels’ class

type

type of decomposition; ie. the reduction is based on

mca meet-complements of atoms in the ‘Pi.rels’ class
pi \(\pi\)-relations in the ‘Pi.rels’ class
at atoms
cc congruence classes

reduc

(optional and logical) does the return object should include the reduced structures?

fac

(optional) the factor that should be decomposed

force

(optional and logical) force further reduction of the semigroup when S has NAs? (see details)

Author

Antonio Rivero Ostoic

Details

The decomp function is a reduction form of an algebraic structure like the semigroup that verifies which of the class members in the system are congruent to each other. The decomposed object then is made of congruent elements, which form part of the lattice of congruence classes in the algebraic structure. In case that the input data comes from the Pacnet program, then such elements are in form of \(\pi\)-relations or the meet-complements of the atoms; otherwise these are simply equivalent elements satisfying the substitution property.

Sometimes a ‘Semigroup’ class object contains not available data in the multiplication table, typically when it is an image from the fact function. In such case, it is possible to perform a reduction of the semigroup structure with the force option, which performs additional equations to the string relations in order to get rid of NAs in the semigroup data.

References

Pattison, Philippa E. Algebraic Models for Social Networks. Cambridge University Press. 1993.

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