In this paper we provide a more general class of non-associative products using the exterior and Clifford bundles on the 7-sphere S 7 . Some additional properties encompass the previous formalisms presented in [1,2] in the Clifford algebra context, and wider classes of non-associative structures on S 7 are investigated, evinced by the directional non-associative products and the mixed composition of generalized non-associative products between Clifford algebra multivectors. These non-associative products are further generalized by considering the non-associative shear of arbitrary Clifford bundle Cℓ0,7 elements into octonions. We assert new properties inherited from the non-associative structure introduced in the whole Clifford bundle on S 7 , which naturally induce involutions on the Clifford bundle and provide immediate generalizations concerning well-established formal results in, e.g., [3,4,5,6,7] and potential applications in physics as pioneered by, e. g., [2,3,4,5,6,7,8,9,10,11,12,13,14].

In this paper we discuss generalized properties of non-associativity in Clifford bundles on the 7-sphere S 7 . Novel and prominent properties inherited from the non-associative structure of the Clifford bundle on S 7 are demonstrated. They naturally lead to general transformations of the spinor fields on S 7 and have dramatic consequences for the associated Kač-Moody current algebras. All additional properties concerning the non-associative structure in the Clifford bundle on S 7 are considered. We further discuss and explore their applications.Mathematics Subject Classification (2010). 15A66, 11E88, 55R25.

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If S is a numerical semigroup and s ∈ S , we denote by nextS(s) = min {x ∈ S | s < x}. Let a be an integer greater than or equal to two. A numerical semigroup is equidistant modulo a if nextS(s) − s − 1 is a multiple of a for every s ∈ S. In this note we give algorithms for computing the whole set of equidistant numerical semigroups modulo a with fixed multiplicity, genus and Frobenius number. Moreover, we will study this kind of semigroups with maximal embedding dimension.

In this work, we study [Formula: see text]-semigroups, that is, numerical semigroups which have no consecutive elements less than the Frobenius number. We give algorithms that allow computation of the whole set of [Formula: see text]-semigroups with a given genus, multiplicity and Frobenius number and from this we study interesting families of [Formula: see text]-semigroups which are Frobenius varieties, pseudo-varieties; R-varieties.