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This paper studies the set of $ntimes n$ matrices for which all row and column sums equal zero. By representing these matrices in a lower dimensional space, it is shown that this set is closed under addition and multiplication, and furthermore is isomorphic to the set of arbitrary $(n-1)times (n-1)$ matrices. The Moore-Penrose pseudoinverse corresponds with the true inverse, (when it exists), in this lower dimension and an explicit representation of this pseudoinverse in terms of the lower dimensional space is given. This analysis is then extended to non-square matrices with all row or all column sums equal to zero.
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Let k be a field. We attach a CW-complex to any Schurian k-category and we prove that the fundamental group of this CW-complex is isomorphic to the intrinsic fundamental group of the k-category. This extends previous results by J.C. Bustamante. We also prove that the Hurewicz morphism from the vector space of abelian characters of the fundamental group to the first Hochschild-Mitchell cohomology vector space of the category is an isomorphism.
In the well-known construction of the field of fractions of an integral domain, division by zero is excluded. We introduce fracpairs as pairs subject to laws consistent with the use of the pair as a fraction, but do not exclude denominators to be zero. We investigate fracpairs over a reduced commutative ring (a commutative ring that has no nonzero nilpotent elements) and provide these with natural definitions for addition, multiplication, and additive and multiplicative inverse. We find that modulo a simple congruence these fracpairs constitute a common meadow, which is a commutative monoid both for addition and multiplication, extended with a weak additive inverse, a multiplicative inverse except for zero, and an additional element a that is the image of the multiplicative inverse on zero and that propagates through all operations. Considering a as an error-value supports the intuition. The equivalence classes of fracpairs thus obtained are called common cancellation fractions (cc-fractions), and cc-fractions over the integers constitute a homomorphic pre-image of the common meadow Qa, the field Q of rational numbers expanded with an a-totalized inverse. Moreover, the initial common meadow is isomorphic to the initial algebra of cc-fractions over the integer numbers. Next, we define canonical term algebras for cc-fractions over the integers and some meadows that model the rational numbers expanded with a totalized inverse, and provide some negative results concerning their associated term rewriting properties. Then we consider reduced commutative rings in which the sum of two squares plus one cannot be a zero divisor: by extending the equivalence relation on fracpairs we obtain an initial algebra that is isomorphic to Qa. Finally, we express negative conjectures concerning alternative specifications for these (concrete) datatypes.
Let V_* be the normalized unitary subgroup of the modular group algebra FG of a finite p-group G over a finite field F with the classical involution *. We investigate the isomorphism problem for the group V_*, that asks when the group V_* is determined by its group algebra FG. We confirm it for classes of finite abelian p-groups, 2-groups of maximal class and non-abelian 2-groups of order at most 16.
Finding necessary and sufficient conditions for isomorphism between two semigroups of order-preserving transformations over an infinite domain with restricted range was an open problem in cite{FHQS}. In this paper, we show a proof strategy to answer that question.
For every profinite group $G$, we construct two covariant functors $Delta_G$ and ${bf {mathcal {AP}}}_G$ from the category of commutative rings with identity to itself, and show that indeed they are equivalent to the functor $W_G$ introduced in [A. Dress and C. Siebeneicher, The Burnside ring of profinite groups and the Witt vectors construction, {it Adv. in Math.} {bf{70}} (1988), 87-132]. We call $Delta_G$ the generalized Burnside-Grothendieck ring functor and ${bf {mathcal {AP}}}_G$ the aperiodic ring functor (associated with $G$). In case $G$ is abelian, we also construct another functor ${bf Ap}_G$ from the category of commutative rings with identity to itself as a generalization of the functor ${bf Ap}$ introduced in [K. Varadarajan, K. Wehrhahn, Aperiodic rings, necklace rings, and Witt vectors, {it Adv. in Math.} {bf 81} (1990), 1-29]. Finally it is shown that there exist $q$-analogues of these functors (i.e, $W_G, Delta_G, {bf {mathcal {AP}}}_G$, and ${bf Ap}_G$) in case $G=hat C$ the profinite completion of the multiplicative infinite cyclic group.
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