In a recent paper by Harada, Seceleanu, and Şega, the Hilbert function, betti table, and graded minimal free resolution of a general principal symmetric ideal are determined when the number of variables in the polynomial ring is sufficiently large. In this paper, we strengthen that result by giving a effective bound on the number of variables needed for their conclusion to hold. The bound is related to a well-known integer sequence involving partition numbers (OEIS A000070). Along the way, we prove a recognition theorem for principal symmetric ideals. We also introduce the class of maximal r-generated submodules, determine their structure, and connect them to general symmetric ideals.

Principal symmetric ideals were recently introduced by Harada, Seceleanu, and Şega, where their homological properties are elucidated. They are ideals generated by the orbit of a single polynomial under permutations of variables in a polynomial ring. In this paper we determine when a product of two principal symmetric ideals is principal symmetric and when the powers of a principal symmetric ideal are again principal symmetric ideals. We characterize the ideals that have the latter property as being generated by polynomials invariant up to a scalar multiple under permutation of variables. Recognizing principal symmetric ideals is an open question for the purpose of which we produce certain obstructions. We also demonstrate that the Hilbert functions of symmetric monomial ideals are not all given by symmetric monomial ideals, in contrast to the non-symmetric case.

It is known that the Sorites paradox can be resolved by working in fuzzy logic and making one of the premises of the argument close to true rather than fully true. We call this premise the Principle of Charity and introduce an additional condition on it, desiring for it to be “almost true” and not just close to true. We proceed to analyze the resolution of the Sorites paradox under various monoidal t-norm based fuzzy logics and determine which of these logics establishes the Principle of Charity as “almost true”. In particular, we desire the Principle of Charity to be “almost true” iff the function mapping objects to their truth values is uniformly continuous. We prove that this relationship holds exactly when the underlying t-norm is nilpotent. This result provides a characterization of both nilpotent t-norms and uniformly continuous functions onto [0, 1].

We introduce and develop the theory of impartial combinatorial games, starting from the basic definitions and working towards a proof of the Sprague-Grundy Theorem, including detailed proofs along the way. Unlike other sources, we consider arbitrary impartial games and not just those with a finite number of moves at each stage. Restricting to the case of impartial games allows the definitions from the general theory of combinatorial games to be simplified for this exposition. This paper was written as part of a capstone course in undergrad.