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==Logical vectors==
{{Group-like structures}}▼
If ''m'' or ''n'' equals one, then the ''m'' × ''n'' logical matrix (M<sub>i j</sub>) 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.
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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''<sup>T</sup> 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's in ''R R''<sup>T</sup> and it fails to be a [[universal relation]].
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