semigroup {multiplex}

Constructing the Semigroup of Relations

Description

Function to create the complete semigroup of multiple relations, where the multiplication table can be specified with either a numerical or a symbolic form.

Usage

semigroup(x, type = c("numerical", "symbolic"), cmps, smpl, valued)

Arguments

x

an array; usually with three dimensions of stacked matrices where the multiple relations are placed

type

semigroup multiplication table to be returned

• numerical for a numerical format (default)

• symbolic for a symbolic format

cmps

(optional and logical) whether the composite matrices should be also given in the output

smpl

(optional and logical) whether to simplify or not the strings of relations

valued

(logical) whether the semigroup should be with a valued format

Details

A multiple relation can be defined by square matrices of 0's and 1's indicating the presence and absence of ties among a set of actors. If there is more than one relation type, the matrices must preserve the label ordering of its elements and stacked into an object array in order to be effectively applied to this function.

The semigroup, which is an algebraic structure having a set with an associative operation on it, is calculated considering binary matrices only. This means that if the provided matrices are valued, the function will dichotomise the input data automatically. Values higher or equal to a unit are converted to one; otherwise they are set to zero. If not happy, use function dichot to specify a cutoff value for the dichotomization.

Semigroup structures for valued relations apply the max min operation in the composition of generators and strings.

Value

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

gens

array with generator relations

cmps

array with the unique compound relations

ord

dimension of the semigroup

st

vector of the unique string relations

S

multiplication matrix with semigroup of relations (see below)

If the specified type is 'numerical', then a matrix of semigroup values is given, otherwise the values is returned as a data frame with the strings of the semigroup.

Warning

For medium size or bigger sets (having e.g. more the 4 relation types), the semigroup construction could take a long time.

Note

It is recommendable to perform the function prev before attempting to construct the semigroup, unless the input data has few dimensions.

Author(s)

Antonio Rivero Ostoic

References

Boorman, S.A. and H.C. White, ‘Social Structure from Multiple Networks. II. Role Structures.’ American Journal of Sociology, 81 (6), 1384-1446. 1976.

Boyd, J.P. Social Semigroups. A unified theory of scaling and blockmodelling as applied to social networks. George Mason University Press. 1991.

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

See Also

green.rel, prev, strings, edgeT, wordT, cngr.

Examples

## create the data: 2 binary relations among 3 elements
arr <- round( replace( array(runif(18), c(3,3,2)), array(runif(18),
       c(3,3,2))>.5, 1 ) )

## optional: put labels
dimnames(arr)[[3]] <- list("n", "m")

## look at the semigroup with numerical format
semigroup(arr)