The first and second Zagreb indices, since its inception have been subjected to an extensive research in the physio- chemical analysis of compounds. In [6] Hanyuan Deng et.al computed the first and second Zagreb indices of four new operations on a graph defined by M. Eliasi, B. Taeri in [4]. Motivated from this we define a new operation on graphs and compute the first and second Zagreb indices of the resultant graph. We illustrate the results with some examples.
第一和第二萨格勒布指数，自成立以来，已受到广泛的研究，在物理化学分析的化合物。在[6]中，邓汉元等人计算了M. Eliasi, B. Taeri在[6]中定义的图上的四种新操作的第一和第二Zagreb指数。在此基础上，我们定义了一个新的图运算，并计算了所得到图的第一和第二萨格勒布指数。我们用一些例子来说明结果。
{"title":"ZAGREB INDICES OF A NEW SUM OF GRAPHS","authors":"Liju Alex, G. Indulal","doi":"10.15826/umj.2023.1.001","DOIUrl":"https://doi.org/10.15826/umj.2023.1.001","url":null,"abstract":"The first and second Zagreb indices, since its inception have been subjected to an extensive research in the physio- chemical analysis of compounds. In [6] Hanyuan Deng et.al computed the first and second Zagreb indices of four new operations on a graph defined by M. Eliasi, B. Taeri in [4]. Motivated from this we define a new operation on graphs and compute the first and second Zagreb indices of the resultant graph. We illustrate the results with some examples.","PeriodicalId":36805,"journal":{"name":"Ural Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43983589","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We develop the first fixed-ratio approximation algorithm for the well-known Prize-Collecting Asymmetric Traveling Salesman Problem, which has numerous valuable applications in operations research. An instance of this problem is given by a complete node- and edge-weighted digraph (G). Each node of the graph (G) can either be visited by the resulting route or skipped, for some penalty, while the arcs of (G) are weighted by non-negative transportation costs that fulfill the triangle inequality constraint. The goal is to find a closed walk that minimizes the total transportation costs augmented by the accumulated penalties. We show that an arbitrary (alpha)-approximation algorithm for the Asymmetric Traveling Salesman Problem induces an ((alpha+1))-approximation for the problem in question. In particular, using the recent ((22+varepsilon))-approximation algorithm of V. Traub and J. Vygen that improves the seminal result of O. Svensson, J. Tarnavski, and L. Végh, we obtain ((23+varepsilon))-approximate solutions for the problem.
{"title":"FIXED RATIO POLYNOMIAL TIME APPROXIMATION ALGORITHM FOR THE PRIZE-COLLECTING ASYMMETRIC TRAVELING SALESMAN PROBLEM","authors":"Ksenia Rizhenko, Katherine Neznakhina, M. Khachay","doi":"10.15826/umj.2023.1.012","DOIUrl":"https://doi.org/10.15826/umj.2023.1.012","url":null,"abstract":"We develop the first fixed-ratio approximation algorithm for the well-known Prize-Collecting Asymmetric Traveling Salesman Problem, which has numerous valuable applications in operations research. An instance of this problem is given by a complete node- and edge-weighted digraph (G). Each node of the graph (G) can either be visited by the resulting route or skipped, for some penalty, while the arcs of (G) are weighted by non-negative transportation costs that fulfill the triangle inequality constraint. The goal is to find a closed walk that minimizes the total transportation costs augmented by the accumulated penalties. We show that an arbitrary (alpha)-approximation algorithm for the Asymmetric Traveling Salesman Problem induces an ((alpha+1))-approximation for the problem in question. In particular, using the recent ((22+varepsilon))-approximation algorithm of V. Traub and J. Vygen that improves the seminal result of O. Svensson, J. Tarnavski, and L. Végh, we obtain ((23+varepsilon))-approximate solutions for the problem.","PeriodicalId":36805,"journal":{"name":"Ural Mathematical Journal","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41698037","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For an arbitrary prime (p), we prove that every algebraic lattice is isomorphic to a complete sublattice in the subgroup lattice of a suitable locally finite (p)-group. In particular, every lattice is embeddable in the subgroup lattice of a locally finite (p)-group.
{"title":"LATTICE UNIVERSALITY OF LOCALLY FINITE (p)-GROUPS","authors":"Vladimir B. Repnitskiǐ","doi":"10.15826/umj.2023.1.011","DOIUrl":"https://doi.org/10.15826/umj.2023.1.011","url":null,"abstract":"For an arbitrary prime (p), we prove that every algebraic lattice is isomorphic to a complete sublattice in the subgroup lattice of a suitable locally finite (p)-group. In particular, every lattice is embeddable in the subgroup lattice of a locally finite (p)-group.","PeriodicalId":36805,"journal":{"name":"Ural Mathematical Journal","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135708518","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The paper deals with a digraph with non-negative vertex weights. A subset (W) of the set of vertices is called dominating if any vertex that not belongs to it is reachable from the set (W) within precisely one step. A dominating set is called minimal if it ceases to be dominating when removing any vertex from it. The paper investigates the problem of searching for a minimal dominating set of maximum weight in a vertex-weighted digraph. An integer linear programming model is proposed for this problem. The model is tested on random instances and the real problem of choosing a family of key indicators in a specific socio-economic system. The paper compares this model with the problem of choosing a dominating set with a fixed number of vertices.
{"title":"THE MINIMAL DOMINATING SETS IN A DIRECTED GRAPH AND THE KEY INDICATORS SET OF SOCIO–ECONOMIC SYSTEM","authors":"R. Simanchev, I. Urazova, V. V. Voroshilov","doi":"10.15826/umj.2023.1.014","DOIUrl":"https://doi.org/10.15826/umj.2023.1.014","url":null,"abstract":"The paper deals with a digraph with non-negative vertex weights. A subset (W) of the set of vertices is called dominating if any vertex that not belongs to it is reachable from the set (W) within precisely one step. A dominating set is called minimal if it ceases to be dominating when removing any vertex from it. The paper investigates the problem of searching for a minimal dominating set of maximum weight in a vertex-weighted digraph. An integer linear programming model is proposed for this problem. The model is tested on random instances and the real problem of choosing a family of key indicators in a specific socio-economic system. The paper compares this model with the problem of choosing a dominating set with a fixed number of vertices.","PeriodicalId":36805,"journal":{"name":"Ural Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43259828","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let (mathcal{D}'(mathbb{R}^n)) and (mathcal{E}'(mathbb{R}^n)) be the spaces of distributions and compactly supported distributions on (mathbb{R}^n), (ngeq 2) respectively, let (mathcal{E}'_{natural}(mathbb{R}^n)) be the space of all radial (invariant under rotations of the space (mathbb{R}^n)) distributions in (mathcal{E}'(mathbb{R}^n)), let(widetilde{T}) be the spherical transform (Fourier–Bessel transform) of a distribution (Tinmathcal{E}'_{natural}(mathbb{R}^n)), and let (mathcal{Z}_{+}(widetilde{T})) be the set of all zeros of an even entire function (widetilde{T}) lying in the half-plane (mathrm{Re} , zgeq 0) and not belonging to the negative part of the imaginary axis. Let (sigma_{r}) be the surface delta function concentrated on the sphere (S_r={xinmathbb{R}^n: |x|=r}). The problem of L. Zalcman on reconstructing a distribution (fin mathcal{D}'(mathbb{R}^n)) from known convolutions (fast sigma_{r_1}) and (fast sigma_{r_2}) is studied. This problem is correctly posed only under the condition (r_1/r_2notin M_n), where (M_n) is the set of all possible ratios of positive zeros of the Bessel function (J_{n/2-1}). The paper shows that if (r_1/r_2notin M_n), then an arbitrary distribution (fin mathcal{D}'(mathbb{R}^n)) can be expanded into an unconditionally convergent series$$f=sumlimits_{lambdainmathcal{Z}_{+}(widetilde{Omega}_{r_1})},,, sumlimits_{muinmathcal{Z}_+(widetilde{Omega}_{r_2})}frac{4lambdamu}{(lambda^2-mu^2) widetilde{Omega}_{r_1}^{,,,displaystyle{'}}(lambda)widetilde{Omega}_{r_2}^{,,,displaystyle{'}}(mu)}Big(P_{r_2} (Delta) big((fastsigma_{r_2})ast Omega_{r_1}^{lambda}big)-P_{r_1} (Delta) big((fastsigma_{r_1})ast Omega_{r_2}^{mu}big)Big)$$in the space (mathcal{D}'(mathbb{R}^n)), where (Delta) is the Laplace operator in (mathbb{R}^n), (P_r) is an explicitly given polynomial of degree ([(n+5)/4]), and (Omega_{r}) and (Omega_{r}^{lambda}) are explicitly constructed radial distributions supported in the ball (|x|leq r). The proof uses the methods of harmonic analysis, as well as the theory of entire and special functions. By a similar technique, it is possible to obtain inversion formulas for other convolution operators with radial distributions.
{"title":"ON ONE ZALCMAN PROBLEM FOR THE MEAN VALUE OPERATOR","authors":"N. Volchkova, V. V. Volchkov","doi":"10.15826/umj.2023.1.017","DOIUrl":"https://doi.org/10.15826/umj.2023.1.017","url":null,"abstract":"Let (mathcal{D}'(mathbb{R}^n)) and (mathcal{E}'(mathbb{R}^n)) be the spaces of distributions and compactly supported distributions on (mathbb{R}^n), (ngeq 2) respectively, let (mathcal{E}'_{natural}(mathbb{R}^n)) be the space of all radial (invariant under rotations of the space (mathbb{R}^n)) distributions in (mathcal{E}'(mathbb{R}^n)), let(widetilde{T}) be the spherical transform (Fourier–Bessel transform) of a distribution (Tinmathcal{E}'_{natural}(mathbb{R}^n)), and let (mathcal{Z}_{+}(widetilde{T})) be the set of all zeros of an even entire function (widetilde{T}) lying in the half-plane (mathrm{Re} , zgeq 0) and not belonging to the negative part of the imaginary axis. Let (sigma_{r}) be the surface delta function concentrated on the sphere (S_r={xinmathbb{R}^n: |x|=r}). The problem of L. Zalcman on reconstructing a distribution (fin mathcal{D}'(mathbb{R}^n)) from known convolutions (fast sigma_{r_1}) and (fast sigma_{r_2}) is studied. This problem is correctly posed only under the condition (r_1/r_2notin M_n), where (M_n) is the set of all possible ratios of positive zeros of the Bessel function (J_{n/2-1}). The paper shows that if (r_1/r_2notin M_n), then an arbitrary distribution (fin mathcal{D}'(mathbb{R}^n)) can be expanded into an unconditionally convergent series$$f=sumlimits_{lambdainmathcal{Z}_{+}(widetilde{Omega}_{r_1})},,, sumlimits_{muinmathcal{Z}_+(widetilde{Omega}_{r_2})}frac{4lambdamu}{(lambda^2-mu^2) widetilde{Omega}_{r_1}^{,,,displaystyle{'}}(lambda)widetilde{Omega}_{r_2}^{,,,displaystyle{'}}(mu)}Big(P_{r_2} (Delta) big((fastsigma_{r_2})ast Omega_{r_1}^{lambda}big)-P_{r_1} (Delta) big((fastsigma_{r_1})ast Omega_{r_2}^{mu}big)Big)$$in the space (mathcal{D}'(mathbb{R}^n)), where (Delta) is the Laplace operator in (mathbb{R}^n), (P_r) is an explicitly given polynomial of degree ([(n+5)/4]), and (Omega_{r}) and (Omega_{r}^{lambda}) are explicitly constructed radial distributions supported in the ball (|x|leq r). The proof uses the methods of harmonic analysis, as well as the theory of entire and special functions. By a similar technique, it is possible to obtain inversion formulas for other convolution operators with radial distributions.","PeriodicalId":36805,"journal":{"name":"Ural Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47676536","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
By an (integer) partition we mean a non-increasing sequence (lambda=(lambda_1, lambda_2, dots)) of non-negative integers that contains a finite number of non-zero components. A partition (lambda) is said to be graphic if there exists a graph (G) such that (lambda = mathrm{dpt},G), where we denote by (mathrm{dpt},G) the degree partition of (G) composed of the degrees of its vertices, taken in non-increasing order and added with zeros. In this paper, we propose to consider another criterion for a partition to be graphic, the ht-criterion, which, in essence, is a convenient and natural reformulation of the well-known Erdös–Gallai criterion for a sequence to be graphical. The ht-criterion fits well into the general study of lattices of integer partitions and is convenient for applications. The paper shows the equivalence of the Gale–Ryser criterion on the realizability of a pair of partitions by bipartite graphs, the ht-criterion and the Erdös–Gallai criterion. New proofs of the Gale–Ryser criterion and the Erdös–Gallai criterion are given. It is also proved that for any graphical partition there exists a realization that is obtained from some splitable graph in a natural way. A number of information of an overview nature is also given on the results previously obtained by the authors which are close in subject matter to those considered in this paper.
{"title":"AROUND THE ERDÖS–GALLAI CRITERION","authors":"V. A. Baransky, T. A. Senchonok","doi":"10.15826/umj.2023.1.003","DOIUrl":"https://doi.org/10.15826/umj.2023.1.003","url":null,"abstract":"By an (integer) partition we mean a non-increasing sequence (lambda=(lambda_1, lambda_2, dots)) of non-negative integers that contains a finite number of non-zero components. A partition (lambda) is said to be graphic if there exists a graph (G) such that (lambda = mathrm{dpt},G), where we denote by (mathrm{dpt},G) the degree partition of (G) composed of the degrees of its vertices, taken in non-increasing order and added with zeros. In this paper, we propose to consider another criterion for a partition to be graphic, the ht-criterion, which, in essence, is a convenient and natural reformulation of the well-known Erdös–Gallai criterion for a sequence to be graphical. The ht-criterion fits well into the general study of lattices of integer partitions and is convenient for applications. The paper shows the equivalence of the Gale–Ryser criterion on the realizability of a pair of partitions by bipartite graphs, the ht-criterion and the Erdös–Gallai criterion. New proofs of the Gale–Ryser criterion and the Erdös–Gallai criterion are given. It is also proved that for any graphical partition there exists a realization that is obtained from some splitable graph in a natural way. A number of information of an overview nature is also given on the results previously obtained by the authors which are close in subject matter to those considered in this paper.","PeriodicalId":36805,"journal":{"name":"Ural Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44452049","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The paper considers the Hyers–Ulam–Rassias stability for systems of nonlinear differential equations with a generalized action on the right-hand side, for example, containing impulses — delta functions. The fact that the derivatives in the equation are considered distributions required a correction of the well known Hyers–Ulam–Rassias definition of stability for such equations. Sufficient conditions are obtained that ensure the property under study.
{"title":"YERS–ULAM–RASSIAS STABILITY OF NONLINEAR DIFFERENTIAL EQUATIONS WITH A GENERALIZED ACTIONS ON THE RIGHT-HAND SIDE","authors":"A. Sesekin, Anna D. Kandrina","doi":"10.15826/umj.2023.1.013","DOIUrl":"https://doi.org/10.15826/umj.2023.1.013","url":null,"abstract":"The paper considers the Hyers–Ulam–Rassias stability for systems of nonlinear differential equations with a generalized action on the right-hand side, for example, containing impulses — delta functions. The fact that the derivatives in the equation are considered distributions required a correction of the well known Hyers–Ulam–Rassias definition of stability for such equations. Sufficient conditions are obtained that ensure the property under study.","PeriodicalId":36805,"journal":{"name":"Ural Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41826687","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper the control system described by a nonlinear differential equation is studied. It is assumed that the control functions have a quadratic integral constraint, more precisely, the admissible control functions are chosen from the ellipsoid of the space (L_2([t_0,theta];mathbb{R}^m)). Different properties of the set of trajectories are investigated. It is proved that a small perturbation of the set of control functions causes also appropriate small perturbation of the set of trajectories. It is also shown that the set of trajectories has a small change if along with the integral constraint on the control functions, a sufficiently large norm type geometric constraint on the control functions is introduced. It is established that every trajectory is robust with respect to the fast consumption of the remaining control resource, and hence every trajectory of the system can be approximated by a trajectory generated by full consumption of the total control resource.
{"title":"ON THE PROPERTIES OF THE SET OF TRAJECTORIES OF THE NONLINEAR CONTROL SYSTEM WITH QUADRATIC INTEGRAL CONSTRAINT ON THE CONTROL FUNCTIONS","authors":"A. Huseyin, N. Huseyin","doi":"10.15826/umj.2023.1.007","DOIUrl":"https://doi.org/10.15826/umj.2023.1.007","url":null,"abstract":"In this paper the control system described by a nonlinear differential equation is studied. It is assumed that the control functions have a quadratic integral constraint, more precisely, the admissible control functions are chosen from the ellipsoid of the space (L_2([t_0,theta];mathbb{R}^m)). Different properties of the set of trajectories are investigated. It is proved that a small perturbation of the set of control functions causes also appropriate small perturbation of the set of trajectories. It is also shown that the set of trajectories has a small change if along with the integral constraint on the control functions, a sufficiently large norm type geometric constraint on the control functions is introduced. It is established that every trajectory is robust with respect to the fast consumption of the remaining control resource, and hence every trajectory of the system can be approximated by a trajectory generated by full consumption of the total control resource.","PeriodicalId":36805,"journal":{"name":"Ural Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47789318","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we consider a class of meromorphic functions (r(z)) having an (s)-fold zero at the origin and establish some inequalities of Bernstein and Turán type for the modulus of the derivative of rational functions in the sup-norm on the disk in the complex plane. These results produce some sharper inequalities while taking into account the placement of zeros of the underlying rational function. Moreover, many inequalities for polynomials and polar derivatives follow as special cases. In particular, our results generalize as well as refine a result due Dewan et al. [6].
{"title":"INEQUALITIES FOR A CLASS OF MEROMORPHIC FUNCTIONS WHOSE ZEROS ARE WITHIN OR OUTSIDE A GIVEN DISK","authors":"Mohd Yousf Mir, S. L. Wali, W. M. Shah","doi":"10.15826/umj.2023.1.008","DOIUrl":"https://doi.org/10.15826/umj.2023.1.008","url":null,"abstract":"In this paper, we consider a class of meromorphic functions (r(z)) having an (s)-fold zero at the origin and establish some inequalities of Bernstein and Turán type for the modulus of the derivative of rational functions in the sup-norm on the disk in the complex plane. These results produce some sharper inequalities while taking into account the placement of zeros of the underlying rational function. Moreover, many inequalities for polynomials and polar derivatives follow as special cases. In particular, our results generalize as well as refine a result due Dewan et al. [6]. ","PeriodicalId":36805,"journal":{"name":"Ural Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46201024","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A non-empty set (S) together with the ternary operation denoted by juxtaposition is said to be ternary semigroup if it satisfies the associativity property (ab(cde)=a(bcd)e=(abc)de) for all (a,b,c,d,ein S). The global set of a ternary semigroup (S) is the set of all non empty subsets of (S) and it is denoted by (P(S)). If (S) is a ternary semigroup then (P(S)) is also a ternary semigroup with a naturally defined ternary multiplication. A natural question arises: "Do all properties of (S) remain the same in (P(S))?" The global determinism problem is a part of this question. A class (K) of ternary semigroups is said to be globally determined if for any two ternary semigroups (S_1) and (S_2) of (K), (P(S_1)cong P(S_2)) implies that (S_1cong S_2). So it is interesting to find the class of ternary semigroups which are globally determined. Here we will study the global determinism of ternary (ast)-band.
非空集合 (S) 与用并置表示的三元运算一起，如果满足结合性，就称为三元半群 (ab(cde)=a(bcd)e=(abc)de) 对所有人 (a,b,c,d,ein S)。三元半群的全局集合 (S) 的所有非空子集的集合是 (S) 用 (P(S))。如果 (S) 那么它是一个三元半群吗 (P(S)) 也是一个具有自然定义的三元乘法的三元半群。一个自然的问题出现了:“所有的属性 (S) 保持不变 (P(S))“全球决定论问题是这个问题的一部分。A类 (K) 对于任意两个三元半群，都说是全局确定的 (S_1) 和 (S_2) 的 (K)， (P(S_1)cong P(S_2)) 这意味着 (S_1cong S_2)。所以找到一类全局确定的三元半群是很有趣的。这里我们将研究三元的全局决定论 (ast)-波段。
{"title":"TERNARY ∗-BANDS ARE GLOBALLY DETERMINED","authors":"Indrani Dutta, S. Kar","doi":"10.15826/umj.2023.1.005","DOIUrl":"https://doi.org/10.15826/umj.2023.1.005","url":null,"abstract":"A non-empty set (S) together with the ternary operation denoted by juxtaposition is said to be ternary semigroup if it satisfies the associativity property (ab(cde)=a(bcd)e=(abc)de) for all (a,b,c,d,ein S). The global set of a ternary semigroup (S) is the set of all non empty subsets of (S) and it is denoted by (P(S)). If (S) is a ternary semigroup then (P(S)) is also a ternary semigroup with a naturally defined ternary multiplication. A natural question arises: \"Do all properties of (S) remain the same in (P(S))?\" The global determinism problem is a part of this question. A class (K) of ternary semigroups is said to be globally determined if for any two ternary semigroups (S_1) and (S_2) of (K), (P(S_1)cong P(S_2)) implies that (S_1cong S_2). So it is interesting to find the class of ternary semigroups which are globally determined. Here we will study the global determinism of ternary (ast)-band.","PeriodicalId":36805,"journal":{"name":"Ural Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48740929","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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