Recent zbMATH articles in MSC 08https://zbmath.org/atom/cc/082021-01-08T12:24:00+00:00WerkzeugDifferent types of cubic ideals in BCI-algebras based on fuzzy points.https://zbmath.org/1449.060302021-01-08T12:24:00+00:00"Jana, Chiranjibe"https://zbmath.org/authors/?q=ai:jana.chiranjibe"Senapati, Tapan"https://zbmath.org/authors/?q=ai:senapati.tapan"Pal, Madhumangal"https://zbmath.org/authors/?q=ai:pal.madhumangal"Borumand Saeid, Arsham"https://zbmath.org/authors/?q=ai:borumand-saeid.arshamSummary: The notions of \((\in ,\in \vee q)\)-cubic \(p\)- (\(a\)- and \(q\)-) ideals of BCI-algebras are introduced and some related properties are investigated. Several characterization for these generalized \((\in ,\in \vee q)\)-cubic ideals are defined and relationship between \((\in ,\in \vee q)\)-cubic \(p\)-ideals, \((\in ,\in \vee q)\)-cubic \(q\)-deals and \((\in ,\in \vee q)\)-cubic \(a\)-ideals of BCI-algebras are discussed.On anti-regular semigroups of \(N (2,2,0)\) algebra.https://zbmath.org/1449.200442021-01-08T12:24:00+00:00"Chen, Lu"https://zbmath.org/authors/?q=ai:chen.luSummary: In the paper, the concept of anti-regular element of \(N (2, 2, 0)\) algebra was introduced. Some examples about regular element, inverse element and anti-regular element were provided, and some properties related to anti-regular semigroup of \(N (2, 2, 0)\) algebra were investigated. Finally, a quotient algebra was constructed by anti-regular element of \(N (2, 2, 0)\) algebra.\(\boldsymbol{X}\)-simplicity of interval max-min matrices.https://zbmath.org/1449.150632021-01-08T12:24:00+00:00"Berežný, Štefan"https://zbmath.org/authors/?q=ai:berezny.stefan"Plavka, Ján"https://zbmath.org/authors/?q=ai:plavka.janSummary: A matrix \(A\) is said to have \(\boldsymbol{X}\)-simple image eigenspace if any eigenvector \(x\) belonging to the interval \(\boldsymbol{X}=\{x:\underline{x}\leq x\leq\overline{x}\}\) containing a constant vector is the unique solution of the system \(A\otimes y=x\) in \(\boldsymbol{X}\). The main result of this paper is an extension of \(\boldsymbol{X}\)-simplicity to interval max-min matrix \(\boldsymbol{A}=\{A:\underline{A}\leq A\leq\overline{A}\}\) distinguishing two possibilities, that at least one matrix or all matrices from a given interval have \(\boldsymbol{X}\)-simple image eigenspace. \(\boldsymbol{X}\)-simplicity of interval matrices in max-min algebra are studied and equivalent conditions for interval matrices which have \(\boldsymbol{X}\)-simple image eigenspace are presented. The characterized property is related to and motivated by the general development of tropical linear algebra and interval analysis, as well as the notions of simple image set and weak robustness (or weak stability) that have been studied in max-min and max-plus algebras.A binary relation on sets of hypersubstitutions for algebraic systems.https://zbmath.org/1449.080012021-01-08T12:24:00+00:00"Phusanga, D."https://zbmath.org/authors/?q=ai:phusanga.daraSummary: Hypersubstitutions for algebraic systems are mappings which send operation symbols to terms and relation symbols to formulas of the corresponding arities. In this paper for any algebraic system \(\mathcal{A}\) we introduce a binary relation \(\sim_{\mathcal{A}}\) on the set \(\mathrm{RelHyp}(\tau, \tau')\) of all hypersubstitutions for algebraic systems of type \( (\tau, \tau')\) which guarantees that for a formula \(F\) of type \( (\tau, \tau')\), the algebraic system \(\mathcal{A}\) satisfies \({\hat \sigma}_1[F]\) if and only if it satisfies \({\hat \sigma}_2[F]\) whenever \({\sigma_1} \sim_{\mathcal{A}}{\sigma_2}\). The aim to introduce \(\sim_{\mathcal{A}}\) is to simplify checking of whether a formula \(F\) is hypersatisfied by an algebraic system \(\mathcal{A}\). It turns out that for solid algebraic systems the relation \(\sim_{\mathcal{A}}\) is a congruence relation on the monoid of all hypersubstitutions of type \( (\tau, \tau')\). This can be generalized to submonoids of the monoid \(\mathrm{RelHyp}(\tau, \tau')\) of all hypersubstitutions of type \( (\tau, \tau')\). For an algebraic system \(\mathcal{A}\) a hypersubstitution is said to be \(\mathcal{A}\)-proper if \(\mathcal{A}\) hypersatisfies all formulas which are satisfied in \(\mathcal{A}\). The set \(P (\mathcal{A})\) of all \(\mathcal{A}\)-proper hypersubstitutions forms a submonoid of the monoid \(\mathrm{RelHyp} (\tau, \tau')\). We prove that \(P (\mathcal{A})\) is saturated with respect to \(\sim_{\mathcal{A}}\).