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Ryszard Urbański - One of the best experts on this subject based on the ideXlab platform.
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commutative semigroups with Cancellation Law a representation theorem
Semigroup Forum, 2011Co-Authors: Jerzy Grzybowski, Diethard Pallaschke, Hubert Przybycien, Ryszard UrbańskiAbstract:Any commutative, cancellative semigroup S with 0 equipped with a uniformity can be embedded in a topological group \(\widetilde{S}\). We introduce the notion of semigroup symmetry T which enables us to turn \(\widetilde{S}\) into an involutive group. In Theorem 2.8 we prove that if S is 2-torsion-free and T is 2-divisible then the decomposition of elements of \(\widetilde{S}\) into a sum of elements of the symmetric subgroup \(\widetilde{S}_{s}\) and the asymmetric subgroup \(\widetilde{S}_{a}\) is polar. In Theorem 3.7 we give conditions under which a topological group \(\widetilde{S}\) is a topological direct sum of its symmetric subgroup \(\widetilde{S}_{s}\) and its asymmetric subgroup \(\widetilde{S}_{a}\). Theorem 2.8 and Theorem 3.7 are designed to be useful tools in studying Minkowski–Radstrom–Hormander spaces (and related topological groups \(\widetilde{S}\)), which are natural extensions of semigroups of bounded closed convex subsets of real Hausdorff topological vector spaces.
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On the Separation and Order Law of Cancellation for Bounded Sets
Optimization, 2002Co-Authors: Diethard Pallaschke, Ryszard UrbańskiAbstract:In this paper we introduce the separation Law for convex sets. Moreover, we prove that for a locally convex topological vector space the order Cancellation Law and separation Law are equivalent.
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On convex class of pairs of convex bodies
Proceedings of the American Mathematical Society, 1997Co-Authors: Jerzy Grzybowski, Ryszard UrbańskiAbstract:In this paper we introduce a quotient class of pairs of convex bodies in which every member have convex union. The space of pairs of convex bodies has been investigated in a number of papers [3], [8], [9], and [12]. This space has found an application in quasidifferential calculus (cf. [1], [5], [7], [10]). A quasidifferential is represented as a pair of convex bodies and it is essential to find the minimal representation of this pair. The notion of minimal pairs was introduced in [5] and investigated in [2], [6], [7] and [11]. Some criteria of minimality are given in [6]. In this paper we investigate pairs of convex bodies with convex union. We introduced a quotient class of pairs of convex compact sets in which every member has convex union. Moreover some criteria for the convex class are given. In this paper X = (X, τ) stands for a real locally convex vector space, and X∗ denotes the dual space of X . Denote by K(X) the family of all convex bodies in X, i.e., of all nonempty compact convex subsets of X . If A,B are nonempty subsets of X , then A+B is the usual algebraic Minkowski sum of A and B. It may be showed that K(X) satisfies the order Cancellation Law; i.e. for every A,B,C ∈ K(X) the inclusion A + B ⊂ B + C implies A ⊂ C (cf. [12]).Hence it follows that K(X) endowed with the Minkowski sum is a commutative semigroup satisfying the Law of Cancellation. Now let K2(X) = K(X)×K(X); the equivalence relation between pairs of convex bodies is given by: (A,B) ∼ (C,D) if and only if A+D = B+C. For A,B ∈ K(X) we will use the notation A∨B := conv (A∪B), where the operation ”conv” denotes the convex hull. If A,B,C ∈ K(X), and b ∈ X , then A∨B+C = (A∨B)+C and A+ b = A+ {b}. We have [a, b] = {a} ∨ {b}. Let f ∈ X∗, A ∈ K(X) and c ∈ R. We denote by pA(f) := maxx∈Af(x) the support function of the set A. Moreover, H f := {x ∈ X | f(x) = c} and HfA := {x ∈ A | f(x) = pA(f)}, where H f is the hyperplane generated by the functional f and the number c, and HfA is the face of A with respect to f. For the sum of the faces of two convex bodies A,B ⊂ X with respect to f ∈ X∗ the identity Hf (A + B) = Hf + HfB holds true. For A ⊂ X we denote by ∂A the boundary Ā \Ao of the set A, where Ā := cl A and A := intA. Received by the editors June 12, 1996. 1991 Mathematics Subject Classification. Primary 52A07, 90C30, 26A27.
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Minimal Fractions of Compact Convex Sets
Variational Analysis and Applications, 1Co-Authors: Diethard Pallaschke, Ryszard UrbańskiAbstract:Pairs of compact convex sets naturally arise in quasidifferential calculus as sub- and super-differentials of a quasidifferentiable function (see [1]). Since the sub- and superdifferential are not uniquely determined, minimal representations are of special importance. In this paper we show that the problem of finding minimal representatives for the elements of pairs of compact convex sets is a special case of the more general problem of determining minimal fractions in ordered commutative semigroups which satisfy the order Cancellation Law. All the material of this paper is taken from the recently published textbook on pairs of compact convex sets ([11]).
Fathiatul Ghina - One of the best experts on this subject based on the ideXlab platform.
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Legality of Testament Cancellation Law and Property Ownership According to Fiqh Al-Syafi’iyyah
Britain International of Humanities and Social Sciences (BIoHS) Journal, 2019Co-Authors: Karimuddin, Khairun Asyura, Syamsul Bahri, Syarkawi, Nurul Husna, Fathiatul GhinaAbstract:Any person who has sufficient assets may inherit a portion of the assets as long as it does not harm the heirs and people who are forced to intend or will not intentionally in their will, the will is invalid. The person who has the will must fulfill the requirements, including adults, sensible, independent and of his own will. So it is not a will made by a minor child and a crazy person. In other cases when the testament inherits the estate and then he cancels the will, or the will inherits more than a third of the total assets but the heir cancels the will, then there will be a problem regarding the legality of the will and the status of ownership of the estate after the Cancellation of the will. Based on these problems, a study is made to find a legal clarity that could be a reference for every policy maker. The results of the study and research can be concluded, al-Syafi'iyyah states that a will is only valid within a third of the inheritance as long as there is no permission from the heirs to testate to more than one third of the assets. exceeds the said level. A will also becomes nullified if a person who has a will cancels his will or inherited property no longer belongs to someone who has a will. Ownership of a will after the will is canceled depends on the reason and the person who cancels it. If the Cancellation is carried out by the willor then the property is returned to the will of the testator, but if the Cancellation of the will is due to a will that exceeds one third of the assets then the will is the right of the heir.
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legality of testament Cancellation Law and property ownership according to fiqh al syafi iyyah
Britain International of Humanities and Social Sciences (BIoHS) Journal, 2019Co-Authors: Khairun Asyura, Syamsul Bahri, Nurul Husna, Fathiatul GhinaAbstract:Any person who has sufficient assets may inherit a portion of the assets as long as it does not harm the heirs and people who are forced to intend or will not intentionally in their will, the will is invalid. The person who has the will must fulfill the requirements, including adults, sensible, independent and of his own will. So it is not a will made by a minor child and a crazy person. In other cases when the testament inherits the estate and then he cancels the will, or the will inherits more than a third of the total assets but the heir cancels the will, then there will be a problem regarding the legality of the will and the status of ownership of the estate after the Cancellation of the will. Based on these problems, a study is made to find a legal clarity that could be a reference for every policy maker. The results of the study and research can be concluded, al-Syafi'iyyah states that a will is only valid within a third of the inheritance as long as there is no permission from the heirs to testate to more than one third of the assets. exceeds the said level. A will also becomes nullified if a person who has a will cancels his will or inherited property no longer belongs to someone who has a will. Ownership of a will after the will is canceled depends on the reason and the person who cancels it. If the Cancellation is carried out by the willor then the property is returned to the will of the testator, but if the Cancellation of the will is due to a will that exceeds one third of the assets then the will is the right of the heir.
H. Michael Damm - One of the best experts on this subject based on the ideXlab platform.
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Half quasigroups and generalized quasigroup orthogonality
Discrete Mathematics, 2011Co-Authors: H. Michael DammAbstract:AbstractIt is still unknown whether three mutually orthogonal Latin squares (resp. quasigroups) of order 10 exist or whether there is a check digit system of order 10 which detects all twin errors. During our research on these topics we use an approach with half quasigroups, which leads to an interesting generalization of quasigroup orthogonality. A (vertical) half quasigroup (H,∗) is a groupoid for which the right Cancellation Law x∗y=x′∗y⇒x=x′ holds. It is close related to what is known as row or column Latin square. The set of all half quasigroups Hn of order n together with an operation ⋅ builds a group (Hn,⋅) and the set of quasigroups Qn is a subset of Hn. Two half quasigroups h,g∈Hn are orthogonal if and only if a quasigroup q∈Qn exists with h⋅q=g. We show that this is just a special case and can be generalized to arbitrary groups.Furthermore, we prove a conjecture of Dénes, Mullen and Suchower about Latin power sets by showing that for all orders n≠2,6 there is a quasigroup q of order n with q2∈Qn and q is orthogonal to q2. Moreover, a computer search verifies a result of Wanless that there is no quasigroup q of order 10 having q2 and q3∈Q10
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Half quasigroups and generalized quasigroup orthogonality
Discrete Mathematics, 2010Co-Authors: H. Michael DammAbstract:It is still unknown whether three mutually orthogonal Latin squares (resp. quasigroups) of order 10 exist or whether there is a check digit system of order 10 which detects all twin errors. During our research on these topics we use an approach with half quasigroups, which leads to an interesting generalization of quasigroup orthogonality. A (vertical) half quasigroup (H,*) is a groupoid for which the right Cancellation Law x*y=x^'*y@?x=x^' holds. It is close related to what is known as row or column Latin square. The set of all half quasigroups H"n of order n together with an operation @? builds a group (H"n,@?) and the set of quasigroups Q"n is a subset of H"n. Two half quasigroups h,g@?H"n are orthogonal if and only if a quasigroup q@?Q"n exists with h@?q=g. We show that this is just a special case and can be generalized to arbitrary groups. Furthermore, we prove a conjecture of Denes, Mullen and Suchower about Latin power sets by showing that for all orders n 2,6 there is a quasigroup q of order n with q^2@?Q"n and q is orthogonal to q^2. Moreover, a computer search verifies a result of Wanless that there is no quasigroup q of order 10 having q^2 and q^3@?Q"1"0.
Diethard Pallaschke - One of the best experts on this subject based on the ideXlab platform.
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commutative semigroups with Cancellation Law a representation theorem
Semigroup Forum, 2011Co-Authors: Jerzy Grzybowski, Diethard Pallaschke, Hubert Przybycien, Ryszard UrbańskiAbstract:Any commutative, cancellative semigroup S with 0 equipped with a uniformity can be embedded in a topological group \(\widetilde{S}\). We introduce the notion of semigroup symmetry T which enables us to turn \(\widetilde{S}\) into an involutive group. In Theorem 2.8 we prove that if S is 2-torsion-free and T is 2-divisible then the decomposition of elements of \(\widetilde{S}\) into a sum of elements of the symmetric subgroup \(\widetilde{S}_{s}\) and the asymmetric subgroup \(\widetilde{S}_{a}\) is polar. In Theorem 3.7 we give conditions under which a topological group \(\widetilde{S}\) is a topological direct sum of its symmetric subgroup \(\widetilde{S}_{s}\) and its asymmetric subgroup \(\widetilde{S}_{a}\). Theorem 2.8 and Theorem 3.7 are designed to be useful tools in studying Minkowski–Radstrom–Hormander spaces (and related topological groups \(\widetilde{S}\)), which are natural extensions of semigroups of bounded closed convex subsets of real Hausdorff topological vector spaces.
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On the Separation and Order Law of Cancellation for Bounded Sets
Optimization, 2002Co-Authors: Diethard Pallaschke, Ryszard UrbańskiAbstract:In this paper we introduce the separation Law for convex sets. Moreover, we prove that for a locally convex topological vector space the order Cancellation Law and separation Law are equivalent.
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Minimal Fractions of Compact Convex Sets
Variational Analysis and Applications, 1Co-Authors: Diethard Pallaschke, Ryszard UrbańskiAbstract:Pairs of compact convex sets naturally arise in quasidifferential calculus as sub- and super-differentials of a quasidifferentiable function (see [1]). Since the sub- and superdifferential are not uniquely determined, minimal representations are of special importance. In this paper we show that the problem of finding minimal representatives for the elements of pairs of compact convex sets is a special case of the more general problem of determining minimal fractions in ordered commutative semigroups which satisfy the order Cancellation Law. All the material of this paper is taken from the recently published textbook on pairs of compact convex sets ([11]).
Khairun Asyura - One of the best experts on this subject based on the ideXlab platform.
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Legality of Testament Cancellation Law and Property Ownership According to Fiqh Al-Syafi’iyyah
Britain International of Humanities and Social Sciences (BIoHS) Journal, 2019Co-Authors: Karimuddin, Khairun Asyura, Syamsul Bahri, Syarkawi, Nurul Husna, Fathiatul GhinaAbstract:Any person who has sufficient assets may inherit a portion of the assets as long as it does not harm the heirs and people who are forced to intend or will not intentionally in their will, the will is invalid. The person who has the will must fulfill the requirements, including adults, sensible, independent and of his own will. So it is not a will made by a minor child and a crazy person. In other cases when the testament inherits the estate and then he cancels the will, or the will inherits more than a third of the total assets but the heir cancels the will, then there will be a problem regarding the legality of the will and the status of ownership of the estate after the Cancellation of the will. Based on these problems, a study is made to find a legal clarity that could be a reference for every policy maker. The results of the study and research can be concluded, al-Syafi'iyyah states that a will is only valid within a third of the inheritance as long as there is no permission from the heirs to testate to more than one third of the assets. exceeds the said level. A will also becomes nullified if a person who has a will cancels his will or inherited property no longer belongs to someone who has a will. Ownership of a will after the will is canceled depends on the reason and the person who cancels it. If the Cancellation is carried out by the willor then the property is returned to the will of the testator, but if the Cancellation of the will is due to a will that exceeds one third of the assets then the will is the right of the heir.
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legality of testament Cancellation Law and property ownership according to fiqh al syafi iyyah
Britain International of Humanities and Social Sciences (BIoHS) Journal, 2019Co-Authors: Khairun Asyura, Syamsul Bahri, Nurul Husna, Fathiatul GhinaAbstract:Any person who has sufficient assets may inherit a portion of the assets as long as it does not harm the heirs and people who are forced to intend or will not intentionally in their will, the will is invalid. The person who has the will must fulfill the requirements, including adults, sensible, independent and of his own will. So it is not a will made by a minor child and a crazy person. In other cases when the testament inherits the estate and then he cancels the will, or the will inherits more than a third of the total assets but the heir cancels the will, then there will be a problem regarding the legality of the will and the status of ownership of the estate after the Cancellation of the will. Based on these problems, a study is made to find a legal clarity that could be a reference for every policy maker. The results of the study and research can be concluded, al-Syafi'iyyah states that a will is only valid within a third of the inheritance as long as there is no permission from the heirs to testate to more than one third of the assets. exceeds the said level. A will also becomes nullified if a person who has a will cancels his will or inherited property no longer belongs to someone who has a will. Ownership of a will after the will is canceled depends on the reason and the person who cancels it. If the Cancellation is carried out by the willor then the property is returned to the will of the testator, but if the Cancellation of the will is due to a will that exceeds one third of the assets then the will is the right of the heir.