Sheaf representations of MV-algebras and lattice-ordered abelian groups via duality

We study representations of MV-algebras -- equivalently, unital lattice-ordered abelian groups -- through the lens of Stone-Priestley duality, using canonical extensions as an essential tool. Specifically, the theory of canonical extensions implies that the (Stone-Priestley) dual spaces of MV-algebras carry the structure of topological partial commutative ordered semigroups. We use this structure to obtain two different decompositions of such spaces, one indexed over the prime MV-spectrum, the other over the maximal MV-spectrum. These decompositions yield sheaf representations of MV-algebras, using a new and purely duality-theoretic result that relates certain sheaf representations of distributive lattices to decompositions of their dual spaces. Importantly, the proofs of the MV-algebraic representation theorems that we obtain in this way are distinguished from the existing work on this topic by the following features: (1) we use only basic algebraic facts about MV-algebras; (2) we show that the two aforementioned sheaf representations are special cases of a common result, with potential for generalizations; and (3) we show that these results are strongly related to the structure of the Stone-Priestley duals of MV-algebras. In addition, using our analysis of these decompositions, we prove that MV-algebras with isomorphic underlying lattices have homeomorphic maximal MV-spectra. This result is an MV-algebraic generalization of a classical theorem by Kaplansky stating that two compact Hausdorff spaces are homeomorphic if, and only if, the lattices of continuous [0, 1]-valued functions on the spaces are isomorphic.

Development of meta-representations: Procedural metacognition and the relationship to Theory of Mind

In several studies it was shown that metacognitive ability is crucial for children and their success in school. Much less is known about the emergence of that ability and its relationship to other meta-representations like Theory of Mind competencies. In the past years, a growing literature has suggested that metacognition and Theory of Mind could theoretically be assumed to belong to the same developmental concept. Since then only a few studies showed empirically evidence that metacognition and Theory of Mind are related. But these studies focused on declarative metacognitive knowledge rather than on procedural metacognitive monitoring like in the present study: N = 159 children were first tested shortly before making the transition to school (aged between 5 1/2 and 7 1/2 years) and one year later at the end of their first grade. Analyses suggest that there is in fact a significant relation between early metacognitive monitoring skills (procedural metacognition) and later Theory of Mind competencies. Notably, language seems to play a crucial role in this relationship. Thus our results bring new insights in the research field of the development of meta-representation and support the view that metacognition and Theory of Mind are indeed interrelated, but the precise mechanisms yet remain unclear.

Explicit description of generic representations for quivers of type An or Dn

We describe explicitly a generic representation for Dynkin quivers of type An or Dn for any dimension vector.