Browsing by Subject "Set theory."

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    Complete intersection monomial curves and non-decreasing Hilbert functions
    (2008) Şahin, Mesut
    In this thesis, we first study the problem of determining set theoretic complete intersection (s.t.c.i.) projective monomial curves. We are also interested in finding the equations of the hypersurfaces on which the monomial curve lie as set theoretic complete intersection. We find these equations for symmetric Arithmetically Cohen-Macaulay monomial curves. We describe a method to produce infinitely many s.t.c.i. monomial curves in P n+1 starting from one single s.t.c.i. monomial curve in P n . Our approach has the side novelty of describing explicitly the equations of hypersurfaces on which these new monomial curves lie as s.t.c.i.. On the other hand, semigroup gluing being one of the most popular techniques of recent research, we develop numerical criteria to determine when these new curves can or cannot be obtained via gluing. Finally, by using the technique of gluing semigroups, we give infinitely many new families of affine monomial curves in arbitrary dimensions with CohenMacaulay tangent cones. This gives rise to large families of 1-dimensional local rings with arbitrary embedding dimensions and having non-decreasing Hilbert functions. We also construct infinitely many affine monomial curves in A n+1 whose tangent cone is not Cohen Macaulay and whose Hilbert function is nondecreasing from a single monomial curve in A n with the same property.
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    Monomial Gotzmann sets
    (2011) Pir, Ata Fırat
    A homogeneous set of monomials in a quotient of the polynomial ring S := F[x1, . . . , xn] is called Gotzmann if the size of this set grows minimally when multiplied with the variables. We note that Gotzmann sets in the quotient R := F[x1, . . . , xn]/(x a 1 ) arise from certain Gotzmann sets in S. Then we partition the monomials in a Gotzmann set in S with respect to the multiplicity of xi and obtain bounds on the size of a component in the partition depending on neighboring components. We show that if the growth of the size of a component is larger than the size of a neighboring component, then this component is a multiple of a Gotzmann set in F[x1, . . . , xi−1, xi+1, . . . xn]. We also adopt some properties of the minimal growth of the Hilbert function in S to R.
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    On the minimal number of elements generating an algebraic set
    (2002) Şahin, Mesut
    In this thesis we present studies on the general problem of finding the minimal number of elements generating an algebraic set in n-space both set and ideal theoretically.
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    Solving equations in the universe of hypersets
    (1993) Pakkan, Müjdat
    Hyperset Theory (a.k.a. ZFC~/AFA) of Peter Aczel is an enrichment of the classical ZFC set theory and uses a graphical representation for sets. By allowing non-well-founded sets, the theory provides an appropriate framework for modeling various phenomena involving circularity. Z F C /A F A has an important consequence that guarantees a solution to a set of equations in the universe of hypersets, viz. the Solution Lemma. This lemma asserts that a system of equations defined in the universe of hypersets has a unique solution, and has applications in areas like artificial intelligence, database theory, and situation theory. In this thesis, a program called HYPERSOLVER, which can solve systems of equations to which the Solution Lemma is applicable and which has built-in procedures to display the graphs depicting the solutions, is presented.