5,248 research outputs found
Classification of small matrices
Denote by the set of square matrices of order . The set
, , is partitioned into row/column permutation equivalence classes
enabling derivation of various facts by simple counting. For example, the
number of regular matrices of order 8 is 10160459763342013440. Let
, denote the set of absolute determinant values and Smith normal
forms of matrices from . Denote by the smallest integer not in
. The sets and are obtained; especially,
. The lower bounds for , , (exceeding the known
lower bound , where is th Fibonacci number) are
obtained. Row/permutation equivalence classes of correspond to bipartite
graphs with black and white vertices, and so the other applications of
the classification are possible.Comment: 45 pages. submitted to LA
Cofrobenius Coalgebra
We investigate left and right co-Frobenius coalgebras and give equivalent
characterizations which prove statements dual to the characterizations of
Frobenius algebras. We prove that a coalgebra is left and right co-Frobenius if
and only if C is isomorphic to Rat(C*) as right C*-modules and also that this
is eqhivalent to the fact that the functors Hom_K(-,K) and Hom_{C*}(-,C*) from
the category of right C-comodules to the category of all left C*-modules are
isomorphic. This allows a definition of a left-right symmetric concept of
co-Frobenius coalgebras that is perfectly dual to the one of Frobenius algebras
and coincides to the existing notion left and right co-Frobenius coalgebra.Comment: 9 pages, to appear in J. Algebr
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