Implementation of the hamming code on pascalabc.net while studying the theoretical foundations of informatics

Theoretical Foundations of Informatics is a classic branch of discrete mathematics taught to students in various information, mathematical and technical fields. The presentation of the material is mainly carried out using matrix algebra. This article describes a methodology for teaching the topic of error correcting coding to pedagogical university students as part of the course on theoretical foundations of informatics and considers an algorithm for obtaining a Hamming code through a tabular template. An original implementation of the considered algorithm in PascalABC. NET is proposed. When writing code, modern techniques are used: dynamic arrays, slices, safe slices, conditional (ternary) operation, foreach loop, sequence methods, lambda expressions, tuples, documenting comments, etc. The functions of the School module from the official delivery of the PascalABC.NET compiler are used to work with binary numbers. The described methodology has been tested in teaching 4-year students of the pedagogical direction, the profile "Mathematics and Informatics" of the Orenburg State Pedagogical University and has shown its applicability. In addition, it is possible to use the proposed approach for teaching schoolchildren at a specialized level or in extracurricular work.

Braided Frobenius algebras from certain Hopf algebras

A braided Frobenius algebra is a Frobenius algebra with a Yang–Baxter operator that commutes with the operations, that are related to diagrams of compact surfaces with boundary expressed as ribbon graphs. A heap is a ternary operation exemplified by a group with the operation [Formula: see text], that is ternary self-distributive. Hopf algebras can be endowed with the algebra version of the heap operation. Using this, we construct braided Frobenius algebras from a class of certain Hopf algebras that admit integrals and cointegrals. For these Hopf algebras we show that the heap operation induces a Yang–Baxter operator on the tensor product, which satisfies the required compatibility conditions. Diagrammatic methods are employed for proving commutativity between Yang–Baxter operators and Frobenius operations.

Ternary semigroups of topological transformations

A ternary semigroup is a nonempty set with a ternary operation which is associative. The purpose of the present paper is to give a characterization of open sets of finite-dimensional Euclidean spaces by ternary semigroups of pairs of homeomorphic transformations and extend to ternary semigroups certain results of L.M. Gluskin concerned with semigroups of homeomorphic transformations of finite-dimensional Euclidean spaces.

Niebrzydowski algebras and trivalent spatial graphs

We introduce Niebrzydowski algebras, algebraic structures with a ternary operation and a partially defined multiplication, with axioms motivated by the Reidemeister moves for [Formula: see text]-oriented trivalent spatial graphs and handlebody-links. As part of this definition, we identify generating sets of [Formula: see text]-oriented Reidemeister moves. We give some examples to demonstrate that the counting invariant can distinguish some [Formula: see text]-oriented trivalent spatial graphs and handlebody-links.

Postulates For Distributive Lattices

Many sets of postulates have been given for distributive lattices and for Boolean algebra. For a description of some of the most interesting and for references to others the reader is referred to Birkhoff's “Lattice Theory”[1]. In this paper we give sets of postulates which have some intrinsic interest because of their simplicity. In the first two sections binary operations are used to describe a distributive lattice by 2 identities in 3 variables and a Boolean algebra by 3 identities in 3 variables. In the third section a ternary operation is used to describe distributive lattices with 0 and J by 2 identities in 5 variables.