Consider the graph on the monoid (under v-multiplication) Fv(R) of divisorial fractional ideals of an integral domain R such that distinct I and J are adjacent if and only if there exist divisorial A and B such that A is covered (under inclusion) in Fv(R) by I and J, and B covers both I and J. The connected component [R] containing R is investigated, and is shown to be a subgroup of Fv(R) for several classes of integral domains, including Krull domains, Prüfer domains, GCD domains, and pseudovaluation domains. In fact, among the main results, it is proved that R is a Krull domain if and only if [R] is a group such that the quotient monoid Fv(R)/[R] is isomorphic to the group of integers. Further characterizations are given for the special cases of Dedekind domains and unique factorization domains.
A semigroup S is called right coherent if every finitely-generated subact of every finitely-presented right S-act is finitely-presented. Left coherency is defined dually, and S is coherent if it is both right and left coherent. In this article we show that graph inverse semigroups are coherent. Such semigroups were introduced by Ash and Hall and are closely connected to C⁎-algebras, Cohn path algebras and Leavitt path algebras.
In this paper, we study prime deductive systems in unital lattice-ordered quantum B-algebras. Assuming the lattice of deductive systems is distributive, we first prove prime extension and intersection representation theorems. We then endow the set P(X) of all proper prime deductive systems with a Zariski topology and investigate its fundamental properties: it is a T0 -space, and under the same distributivity assumption, the subspace M(X) of maximal deductive systems is a T1 -space. In the finite case, we show this space P(X) is spectral. Furthermore, this topology yields a canonical Galois adjunction between the lattice of deductive systems and the lattice of open subsets of P(X), leading to a closure operator that characterizes the spectrally closed deductive systems as its fixed points.
In this paper we develop an approach to the full quaternionic moment problem. We define a hierarchy Hk[q⁎,q]⊂Hk+1[q⁎,q], k∈N0∪{∞}, of two-sided H-linear spaces of quaternionic polynomials which are invariant under conjugation of quaternions and determine the hermitian parts of these spaces explicitly. Using a generalization of Choquet's theorem on adapted spaces to quaternions we provide necessary and sufficient solvability criteria for the quaternionic moment problem of each space Hk[q⁎,q]. The hermitian part of H∞[q⁎,q] is the real polynomial algebra R[x0,x1,x2,x3]. This enables us to apply real algebraic geometry (Positivstellensätze) to the quaternionic moment problem on H∞[q⁎,q].
