Idempotents, group membership and their applications

AbstractŠ. Schwarz in his paper [SCHWARZ, Š.:Zur Theorie der Halbgruppen, Sborník prác Prírodovedeckej fakulty Slovenskej univerzity v Bratislave, Vol. VI, Bratislava, 1943, 64 pp.] proved the existence of maximal subgroups in periodic semigroups and a decade later he brought into play the maximal subsemigroups and thus he embodied the idempotents in the structural description of semigroups [SCHWARZ, Š.:Contribution to the theory of torsion semigroups, Czechoslovak Math. J.3(1) (1953), 7–21]. Later in his papers he showed that a proper description of these structural elements can be used to (re)prove many useful and important results in algebra and number theory. The present paper gives a survey of selected results scattered throughout the literature where an semigroup approach based on tools like idempotent, maximal subgroup or maximal subsemigroup either led to a new insight into the substance of the known results or helped to discover new approach to solve problems. Special attention will be given to some disregarded historical connections between semigroup and ring theory.

Sign patterns of J-orthogonal matrices

This paper builds upon the results in the article “G-matrices, J-orthogonal matrices, and their sign patterns", Czechoslovak Math. J. 66 (2016), 653-670, by Hall and Rozloznik. A number of further general results on the sign patterns of the J-orthogonal matrices are proved. Properties of block diagonal matrices and their sign patterns are examined. It is shown that all 4 × 4 full sign patterns allow J-orthogonality. Important tools in this analysis are Theorem 2.2 on the exchange operator and Theorem 3.2 on the characterization of J-orthogonal matrices in the paper “J-orthogonal matrices: properties and generation", SIAM Review 45 (3) (2003), 504-519, by Higham. As a result, it follows that for n ≤4 all n×n full sign patterns allow a J-orthogonal matrix as well as a G-matrix. In addition, the 3 × 3 sign patterns of the J-orthogonal matrices which have zero entries are characterized.

On approximation by Function having a strong Entropy Point

The paper deals with approximation of functions from the unit interval into itself by means of functions having strong entropy point. For this purpose we define a family of functions having the fixed point property: ConnC (which is a subfamily of the class Conn introduced in [Korczak-Kubiak. E.. Paw- lak. R.J.: Trajectories, first return limiting notions and rings of H-connected and iteratively H-connected functions. Czechoslovak Math. J. 63 (2013). 679-700]). The main result of the paper Is a theorem saying that for any function ƒ ∈ ConnC and any point x0 ∈ Fix(ƒ) there exists a ring R ⊂ ConnC containing function ƒ and in the intersection of any “graph neighbourhood of ƒ” and “neighbourhood of ƒ in topology of uniform convergence”, one can find functions ξ,Ψ ∈ R having a strong entropy point y0 located close to the point x0 and being a discontinuity point of the function ξ and a continuity point of the function Ψ.

On completion in the category SSNσFrm

We introduce the category SSN σ Frm of super strong nearness σ-frames and show the existence of a completion for a super strong nearness σ-frame unique up to isomorphism by the similar construction presented in [WALTERS-WAYLAND, J. L.: Completeness and Nearly Fine Uniform Frames. PhD Thesis, Univ. Catholique de Louvain, 1996] and [WALTERS-WAYLAND, J. L.: A Shirota Theorem for frames, Appl. Categ. Structures 7 (1999), 271–277]. Completion is also shown to be a coreflection in SSN σ Frm. We also engage with the notion of total boundedness for nearness σ-frames and provide a characterization of the Samuel compactification of a nearness σ-frame alternative to the description in [NAIDOO, I.: Samuel compactification and uniform coreflection of nearness σ-frames, Czechoslovak Math. J. 56(131) (2006), 1229–1241].