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Semigroups of lattice points
Scholle, Marek ; Kepka, Tomáš (advisor) ; Šaroch, Jan (referee)
The thesis deals with subsemigroups of (Nm 0 , +), a special discussion is later devoted to the cases m = 1, m = 2 and m = 3. We prove that a subsemigroup of Nm 0 is finitely generated if and only if its generated cone is finitely generated (equivalently polyhedral) and we describe basic topological properties of such cones. We give a few examples illustrating that conditions sufficient for finite generation in N2 0 can not be easily trans- ferred to higher dimensions. We define the Hilbert basis and the related notion of Carathéodory's rank. Besides their basic properties we prove that Carathédory's rank of a subsemigroup of Nm 0 , m = 1, 2, 3, is less than or equal to m. A particular attention is devoted to the subsemigroups containing non-trivial subsemigroups of "subtractive" elements.
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Algebraic Substructures in Cm
Kala, Vítězslav ; Kepka, Tomáš (advisor) ; Stanovský, David (referee) ; El Bashir, Robert (referee)
Title: Algebraic Substructures in ℂ Author: Vítězslav Kala Department: Department of Algebra Supervisor: Prof. RNDr. Tomáš Kepka, DrSc., Department of Algebra Abstract: We study the structure of finitely generated semirings, parasemifields and other algebraic structures, developing and applying tools based on the geom- etry of algebraic substructures of the Euclidean space ℂ . To a parasemifield which is finitely generated as a semiring we attach a certain subsemigroup of the semigroup ℕ0 (defined using elements such that + = for some ∈ and ∈ ℕ). Algebraic and geometric properties of carry important structural information about ; we use them to show that if a parasemifield is 2-generated as a semiring, then it is additively idempotent. We also provide a ring-theoretic reformulation of this conjecture in the case of -generated semirings. We also classify all additively idempotent parasemifields which are finitely gen- erated as semirings by using the fact that they correspond to certain finitely generated unital lattice ordered groups. Busaniche, Cabrer, and Mundici [4] re- cently classified these using the combinatorial and geometric notion of a stellar sequence which is a sequences of certain simplicial complexes in [0, 1] . We use their results to prove that each such parasemifield is a finite product of...
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Finitely generated clones
Draganov, Ondřej ; Barto, Libor (advisor) ; Bulín, Jakub (referee)
A clone is a set of finitary operations closed under composition and contain- ing all projections. We say it is finitely generated if there exist a finite subset {f1, . . . , fn} such that all the other operations can be expressed as compositions of f1, . . . , fn. We present examples of finitely and non-finitely genreated clones on finite sets. First, we demonstrate an explicit construction of operations in finitely generated clones. Secondly, we define relations such that the clones of compatible operations have restricted essential arity, and discuss several modifi- cations. Lastly, for every binary operation f which cannot be composed to yield an essentially ternary operation, we find a maximal clone of essentially at most binary operations containing f. 1
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Algebraic Substructures in Cm
Kala, Vítězslav ; Kepka, Tomáš (advisor) ; Stanovský, David (referee) ; El Bashir, Robert (referee)
Title: Algebraic Substructures in ℂ Author: Vítězslav Kala Department: Department of Algebra Supervisor: Prof. RNDr. Tomáš Kepka, DrSc., Department of Algebra Abstract: We study the structure of finitely generated semirings, parasemifields and other algebraic structures, developing and applying tools based on the geom- etry of algebraic substructures of the Euclidean space ℂ . To a parasemifield which is finitely generated as a semiring we attach a certain subsemigroup of the semigroup ℕ0 (defined using elements such that + = for some ∈ and ∈ ℕ). Algebraic and geometric properties of carry important structural information about ; we use them to show that if a parasemifield is 2-generated as a semiring, then it is additively idempotent. We also provide a ring-theoretic reformulation of this conjecture in the case of -generated semirings. We also classify all additively idempotent parasemifields which are finitely gen- erated as semirings by using the fact that they correspond to certain finitely generated unital lattice ordered groups. Busaniche, Cabrer, and Mundici [4] re- cently classified these using the combinatorial and geometric notion of a stellar sequence which is a sequences of certain simplicial complexes in [0, 1] . We use their results to prove that each such parasemifield is a finite product of...
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Semigroups of lattice points
Scholle, Marek ; Kepka, Tomáš (advisor) ; Šaroch, Jan (referee)
The thesis deals with subsemigroups of (Nm 0 , +), a special discussion is later devoted to the cases m = 1, m = 2 and m = 3. We prove that a subsemigroup of Nm 0 is finitely generated if and only if its generated cone is finitely generated (equivalently polyhedral) and we describe basic topological properties of such cones. We give a few examples illustrating that conditions sufficient for finite generation in N2 0 can not be easily trans- ferred to higher dimensions. We define the Hilbert basis and the related notion of Carathéodory's rank. Besides their basic properties we prove that Carathédory's rank of a subsemigroup of Nm 0 , m = 1, 2, 3, is less than or equal to m. A particular attention is devoted to the subsemigroups containing non-trivial subsemigroups of "subtractive" elements.
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