We shall make a survey of the most recent results obtained in connection with the programme of investigating notable categorical equivalences for MV-algebras from a topos-theoretic perspective commenced in [3]. In [3] and [2] we generalize to a topos-theoretic setting two classical equivalences arising in the context of MV-algebras: Mundici’s equivalence [4] between the category of MV-algebras ...

We give a realization of the crystal B(−∞) for ̂ sl2 using decorated polygons. The construction and proof are combinatorial, making use of Kashiwara and Saito’s characterization of B(−∞), in terms of the ∗ involution. The polygons we use have combinatorial properties suggesting they are the ̂ sl2 analogues of the Mirković-Vilonen polytopes defined by Anderson and the third author in finite type. ...

The varietyMV of all MV-algebras is shown to be non-canonical in a strong sense. Specifically it is shown that the canonical extension of the Chang algebra, K2, is not an MV-algebra. As a consequence, no non-finitely generated variety of MV-algebras is canonical.

For an algebraic group G, Anderson originally defined the notion of MV-polytopes in [And03], images of MV-cycles, defined in [MV07], under the moment map of the corresponding affine Grassmannian. It was shown by Kamnitzer in [Kam07] and [Kam05] that these polytopes can be described via tropical relations and give rise to a crystal structure on the set of MV-cycles. Another crystal structure can...

Let S be a bipartite semigraph with |NXa(y)| ≥ 1 for every y ∈ Y . A vertex x ∈ X hyper dominates y ∈ Y if y ∈ Na(x) or y ∈ Na(NY a(x)). A subsetD ⊆ X is a hyper dominating set of S if every y ∈ Y is hyper dominated by a vertex ofD. A subsetD ⊆ X is called a minimal hyper dominating set of S if no proper subset of D is a hyper dominating set of S. The minimum cardinality of a minimal hyper domi...