Recent zbMATH articles in MSC 20Nhttps://www.zbmath.org/atom/cc/20N2021-04-16T16:22:00+00:00WerkzeugSolution and full classification of generalized binary functional equations of the type \((3;3;0)\).https://www.zbmath.org/1456.390062021-04-16T16:22:00+00:00"Krainichuk, Halyna"https://www.zbmath.org/authors/?q=ai:krainichuk.halyna-v"Sokhatsky, Fedir"https://www.zbmath.org/authors/?q=ai:sokhatsky.fedir-mSummary: Generalized binary functional quasigroup equations in two individual variables with three appearances are under consideration. There exist five classes of the equations (two equations belong to the same class if there exists a relation between sets of their solutions). The quasigroup solution sets of equations from every class are given. In addition, it is proved that every parastrophe of a quasigroup has an orthogonal mate if the quasigroup has an orthogonal mate.Intuitionistic fuzzy relations compatible with the group \(Z_n\).https://www.zbmath.org/1456.030792021-04-16T16:22:00+00:00"Emam, E. G."https://www.zbmath.org/authors/?q=ai:emam.e-gIn the paper the concept of compatibility of fuzzy relations with some multiplicative semi-group is extended to the case of intuitionistic fuzzy relations. The mentioned compatibility of intuitionistic fuzzy relations is studied in the case of the well-known abelian group \(Z_n\) with the sum modulo \(n\) (which is isomorphic to some multiplicative abelian group). As a result the concept of \(Z_n\)-compatiblity of intuitionistic fuzzy relations is considered and several related properties are provided. Namely, a characterization of \(Z_n\)-compatible intuitionistic fuzzy relation is presented, the number of elements in \(Z_n\)-compatible intuitionistic fuzzy relation is determined, preservation of \(Z_n\)-compatiblity by basic operations on intuitionistic fuzzy relations (sum, intersection, converse relation, the complement) as well as some other operations are considered. Moreover, the symmetry of the composition of two \(Z_n\)-compatible intuitionistic fuzzy relations is proved. Finally, two relations related to any intuitionistic fuzzy relation are distinguished and their properties in the context of \(Z_n\)-compatiblity are delivered.
Reviewer: Urszula Bentkowska (Rzeszów)Lie automorphic loops under half-automorphisms.https://www.zbmath.org/1456.200752021-04-16T16:22:00+00:00"Merlini Giuliani, Maria De Lourdes"https://www.zbmath.org/authors/?q=ai:merlini-giuliani.maria-de-lourdes"Dos Anjos, Giliard Souza"https://www.zbmath.org/authors/?q=ai:dos-anjos.giliard-souzaFor a loop \(Q\) denote by \(\operatorname{Inn }Q\) the subgroup in the multplicative group \(\operatorname{Mult }Q\) generated by all permutations
\(R_aR_bR_{ab}^{-1}\), \(L_aL_bL_{bab}^{-1}\), \(R_aL_a^{-1}\) where \(R_q\), \(L_q\) are operators of right and lieft multiplications.
An \(A\)-loop is a loop in which any element from \(\operatorname{Inn }Q\) is an automorphism of \(Q\).
A half-isomorphism \(f:(Q,*)\to (K,\cdot)\) of two groupoids is a bijection such that \(f(x*y)\in \{f(x)\cdot f(y), \, f(y)\cdot f(x)\}\) for all \(x,y\in Q\). If \(Q\) is finite, then for every half-isomorphsim \(f\) of \(Q\) the inverse map \(f^{-1}\) is also a half-isomorphism.
Define in a Lie ring \(L\) a new operation \(x*y=x+y-[x,y]\). Then \((L,*)\) is an \(A\)-loop if and only if \([[x,y],[z,w]]=0\). Let \((L,*)\) be an \(A\)-loop. Then every halp-automorphism or \((L,*)\) is either an automorphism of an anti-isomorphism.
Reviewer: Vyacheslav A. Artamonov (Moskva)Poset loops.https://www.zbmath.org/1456.060032021-04-16T16:22:00+00:00"Smith, J. D. H."https://www.zbmath.org/authors/?q=ai:smith.jonathan-d-hSummary: Given a ring and a locally finite poset, an \textit{incidence loop} or \textit{poset loop} is obtained from a new and natural extended convolution product on the set of functions mapping intervals of the poset to elements of the ring. The paper investigates the interplay between properties of the ring, the poset, and the loop. The annihilation structure of the ring and extremal elements of the poset determine commutative and associative properties for loop elements. Nilpotence of the ring and height restrictions on the poset force the loop to become associative, or even commutative. Constraints on the appearance of nilpotent groups of class 2 as poset loops are given. The main result shows that the incidence loop of a poset of finite height is nilpotent, of nilpotence class bounded in terms of the height of the poset.A characterization of \(n\)-associative, monotone, idempotent functions on an interval that have neutral elements.https://www.zbmath.org/1456.200762021-04-16T16:22:00+00:00"Kiss, Gergely"https://www.zbmath.org/authors/?q=ai:kiss.gergely"Somlai, Gábor"https://www.zbmath.org/authors/?q=ai:somlai.gaborSummary: We investigate monotone idempotent \(n\)-ary semigroups and provide a generalization of the Czogala-Drewniak theorem, which describes the idempotent monotone associative functions having a neutral element. We also present a complete characterization of idempotent monotone \(n\)-associative functions on an interval that have neutral elements.The non-confusing travel groupoids on a finite connected graph.https://www.zbmath.org/1456.050732021-04-16T16:22:00+00:00"Cho, Jung Rae"https://www.zbmath.org/authors/?q=ai:cho.jung-rae"Park, Jeongmi"https://www.zbmath.org/authors/?q=ai:park.jeongmi"Sano, Yoshio"https://www.zbmath.org/authors/?q=ai:sano.yoshioSummary: The notion of travel groupoids was introduced by \textit{L. Nebeský} [Czech. Math. J. 56, No. 2, 659--675 (2006; Zbl 1157.20336)] in connection with a study on geodetic graphs. A travel groupoid is a pair of a set \(V\) and a binary operation \(\ast\) on \(V\) satisfying two axioms. For a travel groupoid, we can associate a graph. We say that a graph \(G\) has a travel groupoid if the graph associated with the travel groupoid is equal to \(G\). A travel groupoid is said to be non-confusing if it has no confusing pairs. Nebeský [loc. cit.] showed that every finite connected graph has at least one non-confusing travel groupoid.
In this note, we study non-confusing travel groupoids on a given finite connected graph and we give a one-to-one correspondence between the set of all non-confusing travel groupoids on a finite connected graph and a combinatorial structure in terms of the given graph.
For the entire collection see [Zbl 1318.68008].