Logical matrix: Difference between revisions

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==Logical vectors==

If ''m'' or ''n'' equals one, then the ''m'' × ''n'' logical matrix (''M''_{i jij}) is a logical vector. If ''m'' = 1, the vector is a row vector, and if ''n'' = 1, it is a column vector. In either case the index equaling one is dropped from denotation of the vector.

Suppose <math>(P_i), \quad i = 1, 2, \ldots, m</math> and <math>(Q_j), \quad j = 1, 2, \ldots, n</math> are two logical vectors. The [[outer product]] of ''P'' and ''Q'' results in an ''m'' × ''n'' [[rectangular relation]]:

:<math>M_{ij} = P_i \land Q_j.</math>

:<math>M_{i j} = P_i \land Q_j .</math> A re-ordering of the rows and columns of such a matrix can assemble all the ones into a rectangular part of the matrix.<ref name=GS>{{cite book | doi=10.1017/CBO9780511778810 | isbn=9780511778810 | author=Gunther Schmidt | page=91 | title=Relational Mathematics | chapter=6: Relations and Vectors | publisher=Cambridge University Press | year=2013 | author-link=Gunther Schmidt }}</ref>

Let ''h'' be the vector of all ones. Then if ''v'' is an arbitrary logical vector, the relation ''R'' = ''v h''^T has constant rows determined by ''v''. In the [[calculus of relations]] such an ''R'' is called a '''vector'''.<ref name=GS/> A particular instance is the universal relation ''h h''^T.

For a given relation ''R'', a maximal, rectangular relation contained in ''R'' is called a '''concept in R'''. Relations may be studied by decomposing into concepts, and then noting the [[heterogeneous relation#Induced concept lattice|induced concept lattice]].

{{Group-like structures}}

Consider the table of group-like structures, where "unneeded" can be denoted 0, and "required" denoted by 1, forming a logical matrix ''R''. To calculate elements of ''R R''^T, it is necessary to use the logical inner product of pairs of logical vectors in rows of this matrix. If this inner product is 0, then the rows are orthogonal. In fact, semigroup is orthogonal to loop, small category is orthogonal to quasigroup, and groupoid is orthogonal to magma. Consequently there are 0'szeros in ''R R''^T, and it fails to be a [[universal relation]].

==Row and column sums==