Pseudo-cohomology of general extensions

Abstract

In this article, we associate new invariants called pseudohomology and pseudo-cohomology groups to right quasigroups and in turn to geometric spaces with natural quasigroup structures and to general extensions. It initiates a program to differentiate different types of general extensions.

Nuclear Properties of Loop Extensions

“…for any α, β, γ ∈ K and a ∈ G. Consequently K × Id f G is left nuclear if and only if (24) is an identity for α = ǫ, hence the assertion (ii) follows. Moreover K × Id f G is a group if and only if it is left nuclear and the identity…”

“…They observed that cohomological cocycles can measure the non-associativity of loop extensions, moreover a cohomological interpretation of the equivalence of extensions was formulated. K. W. Johnson, C. R. Leedham-Green [17], J. D. H. Smith [37], and R. Lal, B. K. Sharma [24] initiated to develop the theory of extensions to more general multiplicative structures. The methods of cohomology theory had successful applications to classification problems of central extensions of abelian groups by loops and to extensions of Bol and Moufang loops by G. P. Nagy, P. Vojtěchovský, D. Daly, [25], [26], [7], N. Nishigôri [32], [33] and R. Jimenez, Q. M. Meléndez [16].…”

On Cocyclic Hadamard Matrices over Goethals-Seidel Loops

“…Yet, surprisingly, no such theory has been fully developed for quasigroups. However, there are some remarkable papers on cohomology for loops (see [16][17][18] and the references therein) and a few for quasigroups [19,20]. For a brief survey on prospective applications of quasigroups and loops, and a panoramic of the state of the art, the interested reader is referred to [21][22][23].…”

“…Lemma 4. Let M ψ be a Goethals-Seidel pseudococyclic matrix, for ψ : GS 4t × GS 4t → {±1} as described in Equation (20), and let i, j ∈ GS 4t . Then, the set of indices k ∈ GS 4t such that σ k (i, jk) = −1 are in one to one correspondence with the ends of the maximal (i, j)-walks (h 0 , .…”