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In finance, optionality is a possible property of a financial contract giving the owner a choice between two or more assets. For example, a convertible bond has optionality because its owner must choose between having a bond or having some shares of stock. In mathematics, a binary operation acts on two elements in a set to produce a third element in that set. When a financial contract such as a convertible bond enjoys optionality between exactly two assets, then the arbitrage-free current value of the contract can potentially be treated as the outcome of a binary operation acting on the two current asset values. In this paper, we treat one of the two assets as riskless and demand that the binary operation linking the two current asset values always produces an arbitrage-free option price. In this context, we focus on the interplay between the properties of the risk-neutral density of the risky asset and the algebraic properties of the binary operation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the notions of the left (right, twosided) ϕ - TL-ideals and (ϕ, σ) - TL-ideals in a semigroup, as well as some of their related properties. Furthermore, the results of relative TLideals are applied to L-ideals of semigroups. [ABSTRACT FROM AUTHOR]

We consider two variants of the sine subtraction law on a semi-group S. The main objective is to solve f(xy∗ ) = f(x)g(y) − g(x)f(y) for unknown functions f, g : S → ℂ, where x ↦ x* is an anti-homomorphic involution. Until now this equation was not solved even when S is a non-Abelian group and x* = x−1. We find the solutions assuming that f is central. A secondary objective is to solve f(xσ(y)) = f(x)g(y) − g(x)f(y), where σ : S → S is a homomorphic involution. Until now this variant was solved assuming that S has an identity element. We also find the continuous solutions of these equations on topological semigroups.

The algebraic extension B Z F of an extended bicyclic semigroup is introduced for an arbitrary ω -closed family ℱ of subsets of ω. It is proved that B Z F is a combinatorial inverse semigroup. We describe Green's relations, the natural partial ordering on the semigroup B Z F , and its set of idempotents. We also establish the criteria of simplicity, 0-simplicity, bisimplicity, and 0-bisimplicity of the semigroup B Z F and a criterion for B Z F to be isomorphic either to the extended bicyclic semigroup or to the countable semigroup of matrix units. It is proved that, in the case where the family ℱ consists of all singletons of ω and the empty set, the semigroup B Z F is isomorphic to the Brandt λ-extension of the semilattice (ω,min). [ABSTRACT FROM AUTHOR]

Our main result is that we describe the solutions g , f : S → C of the functional equation g (x σ (y)) = g (x) g (y) - f (x) f (y) + α f (x σ (y)) , x , y ∈ S , where S is a semigroup, α ∈ C is a fixed constant and σ : S → S an involutive automorphism. [ABSTRACT FROM AUTHOR]

The notion of neutral element generalizes to a pair of elements in ternary algebras. Biunit pairs are introduced as pairs of elements in a semiheap that generalize the notion of Mal'cev element. In order to generalize the known correspondences between semiheaps and certain kinds of semigroups, families of functions generalizing involutions and conjugations, called switches and warps, are investigated. The main theorem establishes that there is a one-to-one correspondence between monoids equipped with a particular switch and semiheaps with a fixed biunit pair. This generalizes the celebrated result in semiheap theory that gives a one-to-one correspondence between involuted monoids and semiheaps with a fixed biunit element. A novel, previously undocumented, algebra is motivated by this result: diheaps are introduced as semiheaps whose elements belong to biunit pairs, which generalize the well-known case of heaps. Diheaps are of great interest since they are shown to be isomorphic to heaps only when they are heaps themselves and explicit non-heap examples are constructed from abelian groups and hypermatrices. [ABSTRACT FROM AUTHOR]

In this paper, we investigate properties, characterizations and compactifications of almost periodic functions with values in a topological vector space. The techniques applied are based essentially on an analogue of a representation in Pettis-integration. Applications of the results to flows are indicated. [ABSTRACT FROM AUTHOR]

Let S be a semigroup. We solve the cosine–sine functional equation f (x y) = f (x) g (y) + g (x) f (y) + h (x) h (y) for unknown functions f , g , h : S → C . The solutions on a group were found by Chung, Kannappan, and Ng in 1985. More recently the solutions on several large classes of semigroups have been found. Here we give the solutions on a general semigroup. The solutions are expressed in terms of multiplicative functions, the solutions of special cases of the sine addition law with one function multiplicative, and the solutions of special cases of the cosine–sine equation with g multiplicative. This gives a complete description since the solutions of the aforementioned special cases are known. The continuous solutions on topological semigroups are also given. [ABSTRACT FROM AUTHOR]

For any semigroup S and any a , b , c ∈ S , the author (Drazin (2012) [3]) defined a as being " (b , c) -invertible", with " (b , c) -inverse" y , if there exists y ∈ S such that y ∈ b S y ∩ y S c (say with b h y = y = y g c), y a b = b and c a y = c. This article is concerned with pairwise implications among y a = a y and other "EP-like" properties of a , b , c , g , h , y such as y c = b y , a h = g a , h y = y g , c a = a b and b h = g c. Each implication is either proved to be always true, or shown by an example to be false. The important special case where b = c and g = h is considered separately, again with a counter-example for every false implication. [ABSTRACT FROM AUTHOR]

In this article, we consider the inclusion ideal graph In(S) of nontrivial right ideals of a semigroup S with zero element. We characterize a semigroup S for which the graph In(S) is complete, connected and also find various graph parameters of In(S). We determine the values of n for which the graph In(Zn) is complete, triangulated, split, unicyclic, thresold and also study minimal embedding of In(Zn) into compact orientable (resp. non-orientable) surface. We give both upper and lower bouds for metric and partition dimension of inclusion ideal graph of a completely 0-simple semigroup. Finally, we compute some graph parameters of the cartesian product of inclusion ideal graph of two monoids. [ABSTRACT FROM AUTHOR]

In this work we study the long time behavior of nonlinear stochastic functional-differential equations in Hilbert spaces. In particular, we start with establishing the existence and uniqueness of mild solutions. We proceed with deriving a priory uniform in time bounds for the solutions in the appropriate Hilbert spaces. These bounds enable us to establish the existence of invariant measure based on Krylov-Bogoliubov theorem on the tightness of the family of measures. Finally, under certain assumptions on nonlinearities, we establish the uniqueness of invariant measures. [ABSTRACT FROM AUTHOR]

We call a semigroup S right noetherian if it satisfies the ascending chain condition on right ideals, and we say that S satisfies ACCPR if it satisfies the ascending chain condition on principal right ideals. We investigate the behavior of these two conditions with respect to ideals and ideal extensions, with a particular focus on minimal and 0-minimal one-sided ideals. In particular, we show that the property of satisfying ACCPR is inherited by right and left ideals. On the other hand, we exhibit an example of a right noetherian semigroup with a minimal ideal that is not right noetherian. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of hybrid bi-ideals in semigroups and investigate some of their important properties. We also give various equivalent conditions for a semigroup to be regular and hybrid structures to be hybrid bi-ideals of S. [ABSTRACT FROM AUTHOR]

Divisibility of elements in a semigroup and the decomposition of a semigroup into subdirect product of semigroups are two basic concepts in the study of the structure of semigroups. The main object of our present paper is to show that the divisibility of elements in a semigroup determines a decomposition of a semigroup of all fuzzy sets in into a subdirect product of semigroups, where the operation on is defined by the following way: for fuzzy set and in , if , and if . [ABSTRACT FROM AUTHOR]

We study the functional equation for unknown functions f , g , h , k , ℓ : S → C , where S is a semigroup and m 1 , m 2 : S → C are multiplicative functions. The study is divided into two main parts: m 1 = m 2 and m 1 ≠ m 2 . In some cases we assume that one or more of the unknown functions is central and/or that S is a monoid. The solutions are found in all cases under the umbrella assumption that S is a commutative monoid. The continuous solutions on topological semigroups are also found. [ABSTRACT FROM AUTHOR]

We study semigroups satisfying the identity a b a = a b . [ABSTRACT FROM AUTHOR]

Let S be a semigroup, let H be a 2 -torsion free abelian group, and let φ , ψ : S → S be two endomorphisms that need not be involutive. We express the solutions f : S → H of generalized variant of Jensen's functional equation \begin{equation*} f(x\varphi(y))+f(\psi(y)x)=2f(x), \quad x,y\in S, \end{equation*} in terms of additive maps. In all results it is supposed that at least one of the endomorphisms φ and ψ is surjective. Many consequences of this result are presented. [ABSTRACT FROM AUTHOR]

We show that a unitary surjective Morita context connecting two semigroups yields Galois connections between certain lattices of compatible relations whenever either semigroup has common weak local units. In the event both semigroups have common weak local units, we obtain mutually inverse lattice isomorphisms that preserve reflexivity, symmetricity and transitivity between the lattices of compatible relations on the semigroups. [ABSTRACT FROM AUTHOR]

In this paper, we study the global well-posedness and exponential stability for a Rao-Nakra sandwich beam equation with time-varying weight and time-varying delay. The system consists of one Euler-Bernoulli beam equation for the transversal displacement, and two wave equations for the longitudinal displacements of the top and bottom layers. By using the semigroup theory, we show that the system is globally well posed. We give two approaches to obtain the exponential stability. The first one is established by multiplier approach provided the coefficients of delay terms are small. We can also obtain the stability by establishing an equivalence between the stabilization of this system and the observability of the corresponding undamped system. The result is new and is the first result of observability on the Rao-Nakra sandwich beam with with time-varying weight and time-varying delay. [ABSTRACT FROM AUTHOR]

This work concerns the study of approximate boundary controllability for some nonlinear partial functional integrodifferential equations with finite delay arising in the modeling of materials with memory, in the framework of general Banach spaces. We give sufficient conditions that ensure the approximate controllability of the system by supposing that its linear undelayed part is approximately controllable, admits a resolvent operator in the sense of Grimmer, and by making use of the Banach fixed-point Theorem and the continuity of the resolvent operator in the uniform norm-topology. As a result, we obtain a generalization of several important results in the literature, without assuming the compactness of the resolvent operator and the uniform boundedness of the nonlinear term. An example of applications is given for illustration. [ABSTRACT FROM AUTHOR]

Discovering maximal frequent itemsets is an important issue and key technique in many data mining problems such as association rule mining. In the literature, generating maximal frequent itemsets proves either to be NP-hard or to have O(l34l(m+n))O\left({l}^{3}{4}^{l}\left(m+n)) complexity in the worst case from the perspective of generating maximal complete bipartite graphs of a bipartite graph, where mm, nn are the item number and the transaction number, respectively, and ll denotes the maximum of ∣C∣∣Ψ(C)∣/(∣C∣+∣Ψ(C)∣−1)| C| | \Psi \left(C)| \hspace{0.1em}\text{/}\hspace{0.1em}\left(| C| +| \Psi \left(C)| -1), with the maximum taken over all maximal frequent itemsets CC. In this article, we put forward a method for discovering maximal frequent itemsets, whose complexity is O(3mn2β+4βn)O\left(3mn{2}^{\beta }+{4}^{\beta }n), lower than the known complexity both in the worst case, from the perspective of semigroup algebra, where β\beta is the number of items whose support is more than the minimum support threshold. Experiments also show that an algorithm based on the algebraic method performs better than the other three well-known algorithms. Meanwhile, we explore some algebraic properties with respect to items and transactions, prove that the maximal frequent itemsets are exactly the simplified generators of frequent itemsets, give a necessary and sufficient condition for a maximal i+1i+1-frequent itemset being a subset of a closed ii-frequent itemset, and provide a recurrence formula of maximal frequent itemsets.

We consider the Levi–Civita equation f(xy)=g1(x)h1(y)+g2(x)h2(y)f\left( {xy} \right) = {g_1}\left( x \right){h_1}\left( y \right) + {g_2}\left( x \right){h_2}\left( y \right) for unknown functions f, g1, g2, h1, h2 : S → ℂ, where S is a monoid. This functional equation contains as special cases many familiar functional equations, including the sine and cosine addition formulas. In a previous paper we solved this equation on groups and on monoids generated by their squares under the assumption that f is central. Here we solve the equation on monoids by two different methods. The first method is elementary and works on a general monoid, assuming only that the function f is central. The second way uses representation theory and assumes that the monoid is commutative. The solutions are found (in both cases) with the help of the recently obtained solution of the sine addition formula on semigroups. We also find the continuous solutions on topological monoids.

It is shown that, in any semigroup with involution, both the DMP inverse of Malik and Thome [Appl Math Comp. 2014;226:575–580.] and the CMP inverse of Mehdipour and Salemi [Linear Multilinear Alg. 2018;66:1046–1053.] are special cases of the Bott–Duffin (e , f) -inverse introduced in [Linear Algebra Appl. 2012;436:1909-1923.] It is also shown that the core inverse of O.M. Baksalary and Trenkler [Linear Multilinear Alg. 2010;58:681–697.] can (even for other pairs of generalized inverses) also be alternatively described, in three other equivalent ways, as the (b , c) -inverse y of a for three other choices of (b , c) besides (b , c) = (a a † , a a †). [ABSTRACT FROM AUTHOR]

We review various constructions of wreath products of groups, semigroups, Lie algebras and associative algebras and discuss their realizations in matrix wreath products of associative algebras. As an application we prove a new version of Evans's embedding theorem [T. Evans, Embedding theorems for multiplicative systems and projective geometries, Proc. American Math. Soc. 3 (1952) 614–620]. [ABSTRACT FROM AUTHOR]

In this paper the concepts of complex fuzzy subsemigroup, complex fuzzy bi-ideal and complex fuzzy ideal in semigroups are introduced and some important characterizations have been obtained. [ABSTRACT FROM AUTHOR]

This paper studies a class of the fractional stochastic evolution equation with delay. With the aid of the compact semigroup theory and Schauder fixed point theorem, the existence of square-mean S-asymptotically periodic mild solutions is obtained under the situation that the nonlinear functions satisfy certain growth conditions. Moreover, by establishing a new Grönwall integral inequality corresponding to fractional differential equation with delay, the global asymptotic stability of the square-mean S-asymptotically periodic mild solutions are discussed. Finally, an example is given to illustrate our abstract results. [ABSTRACT FROM AUTHOR]

In this paper, the focal point is to study the stabilization problem of distributed systems via bilinear boundary control. Hence, we give some sufficient conditions to ensure weak, strong, and exponential stabilization of the considered system. Finally, an example of application is devoted, and the attained results are illustrated through numerical simulations. [ABSTRACT FROM AUTHOR]

Jones monoids n , for 1 < n , is a family of monoids relevant in knot theory. The purpose of this paper is to characterize the identities satisfied by the Jones monoid 5 . [ABSTRACT FROM AUTHOR]

Dai, Hom, Stoffregen and Truong defined a family of concordance invariants φ j . The example of a knot with zero Upsilon invariant but nonzero epsilon invariant previously given by Hom also has nonzero phi invariant. We show there are infinitely many such knots that are linearly independent in the smooth concordance group. In the opposite direction, we build infinite families of linearly independent knots with zero phi invariant but nonzero Upsilon invariant. We also give a recursive formula for the phi invariant of torus knots. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the energy decay of hyperbolic systems of wave-wave, wave-Euler-Bernoulli beam and beam-beam types. The two equations are coupled through boundary connection with only one localized non-smooth fractional Kelvin-Voigt damping. First, we reformulate each system into an augmented model and using a general criteria of Arendt-Batty, we prove that our models are strongly stable. Next, by using frequency domain approach, combined with multiplier technique and some interpolation inequalities, we establish different types of polynomial energy decay rate which depends on the order of the fractional derivative and the type of the damped equation in the system. [ABSTRACT FROM AUTHOR]

The paper studies the properties of a linear operator that is the generator of a semigroup associated with Volterra integro-differential equations in Hilbert spaces. These integro-differential equations can be realized as partial integro-differential equations arising in the theory of viscoelasticity and the theory of heat propagation in media with memory and have a number of other important applications. The results obtained in the paper can be used to study the behavior of and obtain estimates for the solutions of Volterra integro-differential equations. [ABSTRACT FROM AUTHOR]

Hypergraphic automata are automata, state sets and output symbol sets of which are hypergraphs, being invariant under actions of transition and output functions. Universally attracting objects in the category of hypergraphic automata are called universal hypergraphic automata. The semigroups of input symbols of such automata are derivative algebras of mappings for such automata. So their properties are interconnected with the properties of the algebraic structures of the automata. Thus, we can study universal hypergraphic automata by investigating their semigroups of input symbols. Earlier, the authors proved that such automata over hypergraphs from a fairly wide class are completely (up to isomorphism) determined by their semigroups of input symbols. In this paper, we prove the elementary definability of the class of such automata in the class of semigroups. The main result of the paper is the solving of this problem for universal hypergraphic automata over $p$-hypergraphs. It is a wide and very important class of automata because such algebraic systems contain automata whose state hypergraphs and hypergraphs of output symbols are projective or affine planes. The results show that the universal hypergraphic automaton over $p$-hypergraphs is represented as an algebraic system, constructed in the semigroup of input symbols of the automaton using the canonical relations of the automaton. These relations are determined by the formulas of the elementary theory of semigroups. Using such a representation of automata, an effective syntactic transformation of formulas of the elementary theory of hypergraphic automata into formulas of the elementary theory of semigroups is determined. It allows a comprehensive study of the relationship between the elementary properties of universal hypergraphic automata over $p$-hypergraphs and their semigroups of input symbols.

In this work, we study the existence of periodic solutions for a class of linear partial functional differential equations with infinite delay. Inspiring by an existing study, by applying the perturbation theory of semi-Fredholm operators, we introduce a suitable a priori estimate on the norm of the operator L to establish the periodicity of solutions in the case where the linear part is nondensely defined and satisfies the Hille-Yosida condition and without considering the exponential stability condition on the semigroup generated by the part of this operator on the closure of it’s domain. Moreover, in the special case where the linear part generates a strongly continuous semigroup and perturbed by a compact linear operator, we give some sufficient conditions to derive periodic solution from bounded ones. Finally, our theoretical results are illustrated by applications in both densely and nondensely cases.

Models of universe problems are brimming with complexities and uncertainties in almost every field of study, including engineering, mathematics, medical sciences, computer science, physics, management sciences, artificial intelligence, and operations research. To address these uncertainties, various theories have been developed, including probability, rough sets, fuzzy sets, soft ideals, and neutrosophic sets. Neutrosophic set theory is the focus of this paper. In this paper, we introduce the notions of neutrosophic ℵ-filters and neutrosophic ℵ-bi-filters in a semigroup and investigate several properties. Moreover, the relations of prime biideal subset and prime neutrosophic ℵ-bi- ideal structure; neutrosophic ℵ-bi-filter and neutrosophic ℵ-bi-ideal structure; left (resp., right) filter and neutrosophic ℵ-left (resp., right) filter; neutrosophic ℵ-left(resp., right) filter and prime neutrosophic ℵ-left (resp., right) ideals in semigroups are discussed. Finally we prove that: let X be a semigroup and XN be any neutrosophic structure. Then XN is a neutrosophic ℵ-bi-filter of X if and only if XNc is a prime neutrosophic ℵ-bi-ideal of X.

One proves herein that the flow S (t) , generated by the nonlinear Fokker–Planck equation ρ t − Δ β (ρ) + div (a (ρ) ρ) = 0 in (0 , ∞) × R d , is expressed by the Trotter product formula S (t) ρ 0 = lim n → ∞ ⁡ (S A 1 (t n) S A 2 (t n)) n ρ 0 in L 1 (R d) , where S A 1 (t) is the flow (continuous semigroup) generated in L 1 (R d) by the nonlinear diffusion operator A 1 (ρ) = − Δ β (ρ) , while S A 2 (t) is that generated in L 1 (R d) by the conservation law operator A 2 (ρ) = div (a (ρ) ρ) defined in the entropy sense. As an application, one obtains a split-product formula for the McKean–Vlasov stochastic differential equation associated with the Fokker–Planck equation. [ABSTRACT FROM AUTHOR]

The ℛ -height of a semigroup S is the height of the poset of ℛ -classes of S. Given a semigroup S with finite ℛ -height, we establish bounds on the ℛ -height of bi-ideals, one-sided ideals and two-sided ideals; in particular, these substructures inherit the property of having finite ℛ -height. We then investigate whether these bounds can be attained. [ABSTRACT FROM AUTHOR]

In this paper, we consider the stabilization of the generalized Rao‐Nakra beam equation, which consists of four wave equations for the longitudinal displacements and the shear angle of the top and bottom layers and one Euler–Bernoulli beam equation for the transversal displacement. Dissipative mechanism are provided through viscous damping for two displacements. The location of the viscous damping are divided into two groups, characterized by whether both of the top and bottom layers are directly damped or otherwise. Each group consists of three cases. We obtain the necessary and sufficient conditions for the cases in group 2 to be strongly stable. Furthermore, polynomial stability of certain orders are proved. The cases in group 1 are left for future study. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notions of interior filters, quasi-interior filters and weak-interior filters in a quasi-ordered semigroup. Additionally, we study the properties of these types of filters of quasi-ordered semigroups and their interrelationships. [ABSTRACT FROM AUTHOR]

This paper is concerned with the tempered exponential dichotomy for random differential systems in Banach spaces. Based on the presentation of bounded solutions for a tempered exponential dichotomous system, we give a bound under which the tempered exponentially dichotomous property of perturbed systems is preserved. Some applications of our results to stochastic partial differential equations are considered. [ABSTRACT FROM AUTHOR]

In this paper, we consider a viscoelastic plate equation of Moore-Gibson-Thompson viscoelastic plate with heat conduction of Cattaneo type. We investigate the system in bounded and unbounded domains, seeking exponential stability in bounded domains and polynomial decay rates for the Cauchy problem. It turns out that our system is exponentially stable in the bounded domain, while the system has regularity loss in the Cauchy problem. Our results are proved in the sub-critical case K > 0. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a weak form of amenability on topological semigroups that we call φ -amenability, where φ is a character on a topological semigroup. Some basic properties of this new notion are obtained and by giving some examples, we show that this definition is weaker than the amenability of semigroups. As a noticeable result, for a topological semigroup S, it is shown that if S is φ -amenable, then S is amenable. Moreover, φ -ergodicity for a topological semigroup S is introduced and it is proved that under some conditions on S and a Banach space X, φ -amenability and φ -ergodicity of any antirepresntation defined by a right action S on X, are equivalent. A relation between φ -amenability of topological semigroups and the existence of a common fixed point is investigated and by this relation, Hahn-Banach property of topological semigroups in the sense of φ -amenability defined and studied. [ABSTRACT FROM AUTHOR]

In the first part of this paper, a non-autonomous physiologically structured model with nonlocal diffusion is developed. Using the semigroup generated by the diffusion operator and characteristic method, the problem is reformulated as a fixed point problem in a suitable Banach space. We give conditions under which the model admits a unique positive solution. In the second part of this work, we give an application of the study done in the first part. We consider an optimal control problem. The optimal strategies are discussed using normal cone and dual techniques. [ABSTRACT FROM AUTHOR]

A monoid is pseudo-complex if the semigroup variety it generates has uncountably many subvarieties, while the monoid variety it generates has only finitely many subvarieties. The smallest pseudo-complex monoid currently known is of order seven. The present article exhibits a pseudo-complex monoid of order six and shows that every smaller monoid is not pseudo-complex. Consequently, minimal pseudo-complex monoids are of order six. [ABSTRACT FROM AUTHOR]