‎In this paper‎, ‎algebraic investigations on sup-$Sigma$-algebras are presented‎. ‎A representation theorem for‎ ‎sup-$Sigma$-algebras in terms of nuclei and quotients is obtained‎. ‎Consequently‎, ‎the relationship between‎ ‎the congruence lattice of a sup-$Sigma$-algebra and the lattice of its nuclei is fully developed.

For any set representation (permutation representation) of the symmetric group Sn, we give combinatorial interpretation for coefficients of its Frobenius character expanded in the basis of monomial symmetric functions.

We answer a question on the conditioned binomial coefficients raised in and article of Barlotti and Pannone, thus giving an alternative proof of an extension of Frobenius’ generalization of Sylow’s theorem.

The Frobenius theorem is extended from supermanifolds to N-graded manifolds. It is shown, both for super and N-graded manifolds, that the characteristic distribution of a presymplectic submanifold is involutive and hence integrable.

In this paper we show that finite rings for which the code equivalence theorem of MacWilliams is valid for Hamming weight must necessarily be Frobenius. This result makes use of a strategy of Dinh and López-Permouth.