Let X denote a locally connected continuum such that cyclic elements have metrizable boundary in X. We study the cyclic elements of X by demonstrating that each such continuum gives rise to an upper semicontinuous decomposition G of X into continua such that X/G is the continuous image of an arc and the cyclic elements of X correspond to the cyclic elements of X/G that are Peano continua.
We revisit an old question of Knaster by demonstrating that each non-degenerate plane hereditarily unicoherent continuum X contains a proper, non-degenerate subcontinuum which does not separate X.
Let be an integral domain with quotient field . Recall that is Schreier if is integrally closed and for all , implies that where e . A GCD domain is Schreier. We show that an integral domain is a GCD domain if and only if (i) for each pair , there is a finitely generated ideal such that and (ii) every quadratic in that is a product of two linear polynomials in is a product of two linear polynomials in . We also show that is Schreier if and only if every polynomial...
We study the notion of computable categoricity of computable structures, comparing it especially to the notion of relative computable categoricity and its relativizations. We show that every 1 decidable computably categorical structure is relatively Δ⁰₂ categorical. We study the complexity of various index sets associated with computable categoricity and relative computable categoricity. We also introduce and study a variation of relative computable categoricity, comparing it to both computable...
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