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Let P k be the graded polynomial algebra F 2 [ x 1 , x 2 , ... , x k ] over the prime field of two elements, F 2 , with the degree of each x i being 1. We study the hit problem , set up by Frank Peterson, of finding a minimal set of generators for P k as a module over the mod-2 Steenrod algebra, A. In this paper, we explicitly determine a minimal set of A -generators for P 5 in the case of the generic degrees n = 2 d + 1 − 1 and n = 2 d + 1 − 2 for all d ⩾ 6. [ABSTRACT FROM AUTHOR]

A Boolean algebra A equipped with a (finitely-additive) positive probability measure m can be turned into a metric space (A , d m) , where d m (a , b) = m ((a ∧ ¬ b) ∨ (¬ a ∧ b)) , for any a , b ∈ A , sometimes referred to as metric Boolean algebra. In this paper, we study under which conditions the space of atoms of a finite metric Boolean algebra can be isometrically embedded in R N (for a certain N) equipped with the Euclidean metric. In particular, we characterize the topology of the positive measures over a finite algebra A such that the metric space (At (A) , d m) embeds isometrically in R N (with the Euclidean metric). [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the Nijenhuis operators on Hom-Lie conformal algebras. We construct a graded Lie algebra whose Maurer-Cartan elements are given by Nijenhuis operators with the aid of Frölicher-Nijenhuis bracket. After constructing graded Lie algebra, we define the cohomology associated with a Nijenhuis operator. We further introduce Hom- NS Lie conformal algebra as an algebraic structure behind Nijenhuis operators on Hom-Lie conformal algebra. We provide various examples of Hom- NS Lie conformal algebras. Finally, as an application to cohomology, we study formal deformations of Nijenhuis operators. [ABSTRACT FROM AUTHOR]

Under a general categorical procedure for the extension of dual equivalences as presented in this paper's predecessor, a new algebraically defined category is established that is dually equivalent to the category LKHaus of locally compact Hausdorff spaces and continuous maps, with the dual equivalence extending a Stone-type duality for the category of extremally disconnected locally compact Hausdorff spaces and continuous maps. The new category is then shown to be isomorphic to the category CLCA of complete local contact algebras and suitable morphisms. Thereby, a new proof is presented for the equivalence LKHaus ≃ CLCA op that was obtained by the first author more than a decade ago. Unlike the morphisms of CLCA , the morphisms of the new category and their composition law are very natural and easy to handle. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to compute all possible mod 2 cohomology algebra of orbit spaces of free involutions on odd dimensional Wall manifolds of type Q (m , n) , which are the closed differentiable manifolds constructed as mapping torus of specific involutions on Dold manifolds P (m , n) , for m even and n odd. [ABSTRACT FROM AUTHOR]

The aim of this paper is to show that Scott-closed sets have a good application in convex structures. Firstly, based on remotehood system, we give a characterization of convex structure by the Kleisli monoid with respect to the Scott-closed set monad. Then, by an op-canonical lax extension of the Scott-closed set monad, we introduce convex convergence spaces and prove that convex structures are precisely the reflexive and transitive lax algebras. Finally, we study the relationship between ordered structures and T 0 separated convex structures. [ABSTRACT FROM AUTHOR]

We write P for the polynomial algebra in one variable over the finite field Z 2 and P ⊗ t = Z 2 [ x 1 , ... , x t ] for its t -fold tensor product with itself. We grade P ⊗ t by assigning degree 1 to each generator. We are interested in determining a minimal set of generators for the ring of invariants (P ⊗ t) G t as a module over Steenrod ring, A 2. Here G t is a subgroup of the general linear group G L (t , Z 2). An equivalent problem is to find a monomial basis of the space of "unhit" elements, Z 2 ⊗ A 2 (P ⊗ t) G t in each t and degree n ≥ 0. The structure of this tensor product is proved surprisingly difficult and has been not yet known for t ≥ 5 , even for the trivial subgroup G t = { e }. In the present paper, we consider the subgroup G t = { e } for t ∈ { 5 , 6 } , and obtain some new results on A 2 -generators of (P ⊗ t) G t in some degrees. At the same time, some of their applications have been proposed. We also provide an algorithm in MAGMA for verifying the results. This study can be understood as a continuation of our recent works in [24,26]. [ABSTRACT FROM AUTHOR]

A multiple conjugation quandle is an algebra whose axioms are motivated from handlebody-knot theory. Any linear extension of a multiple conjugation quandle can be described by using a pair of maps called an MCQ Alexander pair. In this paper, we show that any affine extension of a multiple conjugation quandle can be described by using a quadruple of maps, called an augmented MCQ Alexander pair. [ABSTRACT FROM AUTHOR]

A quandle is an algebra with two binary operations satisfying three conditions which are related to Reidemeister moves in knot theory. In this paper we introduce the notion of the (canonical) tensor product of a quandle. The tensor product of the knot quandle or the knot symmetric quandle of a surface-link in 4-space can be used to classify or construct invariants of 1-handles attaching to the surface-link. We also compute the tensor products for dihedral quandles and their symmetric doubles. [ABSTRACT FROM AUTHOR]

Biquandles and multiple conjugation biquandles are algebras which are related to links and handlebody-links in 3-space. Cocycles of biquandles and multiple conjugation quandles can be used to construct state-sum type invariants of links and handlebody-links. In this paper we discuss cocycles of a certain class of biquandles and multiple conjugation biquandles, which we call G -Alexander biquandles and G -Alexander multiple conjugation biquandles, with a relationship with group cocycles. We give a method to obtain a (biquandle or multiple conjugation biquandle) cocycle from a group cocycle. [ABSTRACT FROM AUTHOR]