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Nullnorms with a zero element being at any point of a bounded lattice are an important generalization of triangular norms and triangular conorms. This paper obtains an equivalent characterization for the existence of idempotent nullnorms with the zero element a on any bounded lattice containing only two distinct elements incomparable with a. Furthermore, some basic properties for the bounded lattice containing only two distinct elements incomparable with a are presented.

Demonic composition, demonic refinement and demonic union are alternatives to the usual ‘angelic’ composition, angelic refinement (inclusion) and angelic (usual) union defined on binary relations. We first motivate both the angelic and the demonic via an analysis of the behaviour of non-deterministic programs, with the angelic associated with partial correctness and demonic with total correctness, both cases emerging from a richer algebraic model of non-deterministic programs incorporating both aspects. Zareckiĭ has shown that the isomorphism class of algebras of binary relations under angelic composition and inclusion is finitely axiomatized as the class of ordered semigroups. The proof can be used to establish that the same axiomatization applies to binary relations under demonic composition and refinement, and a further modification of the proof can be used to incorporate a zero element representing the empty relation in the angelic case and the full relation in the demonic case. For the signature of angelic composition and union, it is known that no finite axiomatization exists, and we show the analogous result for demonic composition and demonic union by showing that the same axiomatization holds for both. We show that the isomorphism class of algebras of binary relations with the ‘mixed’ signature of demonic composition and angelic inclusion has no finite axiomatization. As a contrast, we show that the isomorphism class of partial algebras of binary relations with the partial operation of constellation product and inclusion (also a ‘mixed’ signature) is finitely axiomatizable.

We introduce a new class of semigroups, that we call BZS - Boolean Zero Square-semigroups. A semigroup S with a zero element, 0, is said to be a BZS semigroup if, for every , we have or . We obtain some properties that describe the behaviour of the Green’s equivalence relations , , and . Necessary and sufficient conditions for a BZS semigroup to be a band and an inverse semigroup are obtained. A characterisation of a special type of BZS completely 0-simple semigroup is presented.

Let $\mathcal C_{n}$ be the Catalan monoid on $X_{n}=\{1,\ldots ,n\}$ under its natural order. In this paper, we describe the sets of left zero-divisors, right zero-divisors and two sided zero-divisors of $\mathcal C_{n}$; and their numbers. For $n \geq 4$, we define an undirected graph $\Gamma(\mathcal C_{n})$ associated with $\mathcal C_{n}$ whose vertices are the two sided zero-divisors of $\mathcal C_{n}$ excluding the zero element $\theta$ of $\mathcal C_{n}$ with distinct two vertices $\alpha$ and $\beta$ joined by an edge in case $\alpha\beta=\theta=\beta\alpha$. Then we first prove that $\Gamma(\mathcal C_{n})$ is a connected graph, and then we find the diameter, radius, girth, domination number, clique number and chromatic numbers and the degrees of all vertices of $\Gamma(\mathcal C_{n})$. Moreover, we prove that $\Gamma(\mathcal C_{n})$ is a chordal graph, and so a perfect graph.

The usual approximation solutions with variable ordering structures fail in some cases such as Examples 2.1 and 2.2, where the intersection of all ordering cones equals to the zero element. Inspire...

Let R denote a commutative Noetherian ring and I an ideal of R. The concept of quintessential sequences over zero ideal was introduced by McAdam and Ratliff (1985) [8] . They showed that these sequences enjoy many of the basic properties of asymptotic sequences over zero ideal. It was shown, that quintessential sequences over ideals I ≠ ( 0 ) R are not a good analogue of asymptotic sequences over I ≠ ( 0 ) R . By making use of the new concept of quintessential grade of an ideal over another ideal, we show that there exists a class of ideals I for which quintessential sequences over I are an excellent analogue of asymptotic sequences over I. Also, we give more results on quintessential sequences over an ideal and derive generalizations of some McAdam-Ratliff's results (1985) [8] , [13] .

An ideal $I$ of a ring $R$ is called left N-reflexive if for any $a\in$ nil$(R)$, $b\in R$, being $aRb \subseteq I$ implies $bRa \subseteq I$ where nil$(R)$ is the set of all nilpotent elements of $R$. The ring $R$ is called left N-reflexive if the zero ideal is left N-reflexive. We study the properties of left N-reflexive rings and related concepts. Since reflexive rings and reduced rings are left N-reflexive, we investigate the sufficient conditions for left N-reflexive rings to be reflexive and reduced. We first consider basic extensions of left N-reflexive rings. For an ideal-symmetric ideal $I$ of a ring $R$, $R/I$ is left N-reflexive. If an ideal $I$ of a ring $R$ is reduced as a ring without identity and $R/I$ is left N-reflexive, then $R$ is left N-reflexive. If $R$ is a quasi-Armendariz ring and the coefficients of any nilpotent polynomial in $R[x]$ are nilpotent in $R$, it is proved that $R$ is left N-reflexive if and only if $R[x]$ is left N-reflexive. We show that the concept of N-reflexivity is weaker than that of reflexivity and stronger than that of left N-right idempotent reflexivity and right idempotent reflexivity which are introduced in Section 5.

A multiplicative subset of a commutative ring contains the zero element precisely if the set in question meets every prime ideal. While this form of Krull's Lemma takes recourse to transfinite reasoning, it has recently allowed for a crucial reduction to the integral case in Kemper and the third author's novel characterization of the valuative dimension. We present a dynamical solution by which transfinite reasoning can be avoided, and illustrate this constructive method with concrete examples. We further give a combinatorial explanation by relating the Zariski lattice to a certain inductively generated class of finite binary trees. In particular, we make explicit the computational content of Krull's Lemma.

A zero ring is a ring in which the product of any two elements is zero, which is the additive identity. A zero ring labeling of a graph is an assignment of distinct elements of a zero ring to the vertices of the graph such that the sum of the labels of any two adjacent vertices is not the zero element in the ring. Given a zero ring labeling of a graph, if the cardinality of the set of distinct sums obtained from all adjacent vertices is equal to the maximum degree of the graph, then the zero ring labeling is efficient. In this paper, we showed the existence of an efficient zero ring labeling for some classes of trees and their disjoint union. In particular, we showed that an efficient zero ring labeling exists for some families of the following classes of trees: path graphs, star graphs, bistars, centipede graphs, caterpillars, spiders, lobsters, and rooted trees. We also showed results for other common classes of graphs.

Nullnorms as a generalization of the concepts of t-norms and t-conorms admit a zero element to be any element in a bounded lattice. In this paper, we discuss the characterization of idempotent nullnorms on bounded lattices and investigate some of the main properties of them. We propose some new methods yielding nullnorms, in particular idempotent nullnorms, on bounded lattices with the fixed zero element under some additional constraints. Moreover, some illustrative examples are presented to demonstrate the feasibility of the proposed methods.

In the symmetric key cryptography, the purpose of the substitution box is to generate confusion and hence improve the security of the whole cryptosystem. For this reason, many algebraic substitution boxes are constructed depending upon the associative algebras of Galois fields. In this paper, the power associative loop is used for the construction of substitution boxes. This novel structure comprises excellent features which include, the inverse of zero element, non-associativity and fewer constraints as compared to the cyclic group and Galois field. In comparison to existing substitution boxes, the substitution boxes based upon power associative loop are comparatively easy and the above-mentioned properties offer the number of structures to construct highly nonlinear substitution boxes. To obtain the number of substitution boxes, we further applied the symmetric group of order 16 on the proposed substitution box. The evaluation of proposed boxes with different algebraic and statistical analyses like nonlinearity test, strict avalanche criterion, bit independence criterion, linear approximation probability and differential approximation probability indicate the strength of proposed substitution boxes. Majority logic criterion results depict that proposed substitution boxes have better cryptographic strength to apply in different techniques of secure communication.

Discrete tori are $${\mathbb {Z}}_m^N$$ thought of as vertices of graphs $${\mathcal {C}}_m^N$$ whose adjacencies encode the Cartesian product structure. Space-limiting refers to truncation to a symmetric path neighborhood of the zero element and spectrum-limiting in this case refers to corresponding truncation in the isomorphic Fourier domain. Composition spatio-spectral limiting (SSL) operators are analogues of classical time and band limiting operators. Certain adjacency-invariant spaces of vectors defined on $${\mathbb {Z}}_m^N$$ are shown to have bases consisting of Fourier transforms of eigenvectors of SSL operators. We show that when $$m=3$$ or $$m=4$$ , all eigenvectors of SSL arise in this way. We study the structure of corresponding invariant spaces when $$m\ge 5$$ and give an example to indicate that the relationship between eigenvectors of SSL and the corresponding adjacency-invariant spaces should extend to $$m\ge 5$$ .

In block ciphers, the substitution boxes (S-boxes) are used with the purpose to induce confusion in cryptosystems. For the last three decades most of algebraic and chaotic S-boxes are based on the associative algebras of Galois fields. However, recently a little but remarkable work on the designing of S-boxes over some local associative algebras have been published, which is in fact a mock of the associative algebras of Galois fields. The main purpose of these establishments is to hide data and improve the security levels of crypto algorithms. In this study we redesign S-boxes over the elements of inverse property loop. The fascinating features of this structure are; it’s non-associativity and the existence of the inverse of zero element. These properties not only increase the availability of the number of structures but also provide easy approach in constructing S-boxes. The newly designed S-boxes are assessed by various analyses, which include nonlinearity, strict avalanche criterion, bit independence criterion, linear approximation probabilities, and differential approximation probabilities. Majority logic criterion is also performed to evaluate its application in various encryption systems.

The main result of the paper is the classification of all (nonassociative) algebras of level two, i.e. such algebras that maximal chains of nontrivial degenerations starting at them have length two. During this classification we obtain an estimation of the level of an algebra via its generation type, i.e. the maximal dimension of its one generated subalgebra. Also we describe all degenerations and levels of algebras of the generation type $1$ with a square zero ideal of codimension $1$.

We classify plethories over fields of characteristic zero, thus answering a question of Borger-Wieland and Bergman-Hausknecht. All plethories over characteristic zero fields are linear, in the sense that they are free plethories on a bialgebra. For the proof we need some facts from the theory of ring schemes where we extend previously known results. We also classify plethories with trivial Verschiebung over a perfect field of non-zero characteristic and indicate future work., 17 pages, comments welcome

This paper propose an acceleration method of Compressed Sensing by both Kronecker product and zero element removal in estimation of both DOA and TOA. Numerical results show that the calculation time is greatly reduced is by proposed method.

In this study, we define new semigroup structures using the set S S = a ∈ S | a S a = 0 which is called the source of semiprimeness for a semigroup S with zero element. S S − idempotent semigroup, S S − regular semigroup, S S − reduced semigroup, and S S − nonzero divisor semigroup which are generalizations of idempotent, regular, reduced, and nonzero divisor semigroups in semigroup theory are investigated, and their basic properties are determined. In addition, we adapt some well-known results in semigroup theory to these new semigroups.

In this chapter, we study certain graphs obtained from total graphs of commutative rings. More specifically, we concentrate on the total graph without the zero element, the complement of the total graph, and its generalizations.

As a generalization of the concepts of t-norm and t-conorm on a bounded lattice, nullnorms are important from a theoretical viewpoint but also because of their numerous potential applications, such as in expert systems, fuzzy quantifiers and neural networks. Nullnorms have a zero element selected from the whole domain, regardless of whether t-norms and t-conorms have a zero element on the lower and upper elements of a bounded lattice. There are several attempts to investigate the structure of nullnorms on bounded lattices in literature. In this paper, unlike the researches in the literature, we use the closure operators and interior operators on a bounded lattice to construct new classes of nullnorms with a zero element under some additional conditions. Furthermore, several interesting examples are included to have a better understanding of new structures of nullnorms on bounded lattices.

Let $\mathscr{F}=(F_n)$ be a sequence of nonempty finite subsets of $\omega$ such that $\lim_n |F_n|=\infty$ and define the ideal $$\mathcal{I}(\mathscr{F}):=\left\{A\subseteq \omega: |A\cap F_n|/|F_n|\to 0~\mbox{as}~n\to \infty \right\}.$$ The case $F_n=\{1,\ldots,n\}$ corresponds to the classical case of density zero ideal. We show that $\mathcal{I}(\mathscr{F})$ is an analytic P-ideal but not $F_{\sigma}$. As a consequence, we show that the set of real bounded sequences which are $\mathcal{I}(\mathscr{F})$-convergent to $0$ is not complemented in $\ell_\infty$., Comment: 3 pages

In this paper, we introduce two rather effective methods for constructing new families of nullnorms with a zero element on the basis of the closure operators and interior operators on a bounded lattice under some additional conditions. Our constructions can be seen as a generalization of the ones in [28]. As a by-product, two types of idempotent nullnorms on bounded lattices are obtained. Several interesting examples are included to get a better understanding of the structure of new families of nullnorms.

Based on the natural vector addition and scalar multiplication, the set of all bounded and closed intervals in R cannot form a vector space. This is mainly because the zero element does not exist. In this paper, we endow a norm to the interval space in which the axioms are almost the same as the axioms of conventional norm by involving the concept of null set. Under this consideration, we shall propose two different concepts of open balls. Based on the open balls, we shall also propose the different types of open sets, which can generate many different topologies.

A charge (finitely additive measure) defined on a Boolean algebra of sets taking values in a group G G is called a strictly nonzero (SNZ) charge if it takes the identity value in G G only for the zero element of the Boolean algebra. A study of such charges was initiated by Rüdiger Göbel and K. P. S. Bhaskara Rao in 2002. Our main result is a solution to one of the questions posed in that paper: we show that for every cardinal ℵ \aleph , the Boolean algebra of clopen sets of { 0 , 1 } ℵ \{0,1\}^\aleph has a strictly nonzero integer-valued charge. The key lemma that we prove is that there exists a strictly nonzero integer-valued permutation-invariant charge on the Boolean algebra of clopen sets of { 0 , 1 } ℵ 0 \{0,1\}^{\aleph _0} . Our proof is based on linear-algebraic arguments, as well as certain kinds of polynomial approximations of binomial coefficients. We also show that there is no integer-valued SNZ charge on P ( N ) {\mathcal {P}}(\mathbb {N}) . Finally, we raise some interesting problems on integer-valued SNZ charges.