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You can have non-binary logic matrices too. It's a symmetric closed monoidal category instead of a closed Cartesian one |
m →Logical vectors: correct for 5 column table |
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==Logical vectors==
{{Group-like structures}}▼
If ''m'' or ''n'' equals one, then the ''m'' × ''n'' logical matrix (''m''<sub>''ij''</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 1 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]].
Consider the table of group-like structures, where "unneeded" can be denoted 0, and "required" denoted by 1, forming a logical matrix
▲{{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 <math>RR^{\operatorname{T}}</math>, 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 (algebra)|loop]], [[small category]] is orthogonal to [[quasigroup]], and [[groupoid]] is orthogonal to [[magma]]. Consequently there are zeros in <math>RR^{\operatorname{T}}</math>, and it fails to be a [[universal relation]].
==Row and column sums==
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