We extend the theory of unified correspondence to a broad class of logics with algebraic semantics given by varieties of normal lattice expansions (LEs), also known as ‘lattices with operators’. ...Specifically, we introduce a syntactic definition of the class of Sahlqvist formulas and inequalities which applies uniformly to each LE-signature and is given purely in terms of the order-theoretic properties of the algebraic interpretations of the logical connectives. We also introduce the algorithm ALBA, parametric in each LE-setting, which effectively computes first-order correspondents of LE-inequalities, and is guaranteed to succeed on a wide class of inequalities (the so-called inductive inequalities) which significantly extend the Sahlqvist class. Further, we show that every inequality on which ALBA succeeds is canonical. Projecting these results on specific signatures yields state-of-the-art correspondence and canonicity theory for many well known modal expansions of classical and intuitionistic logic and for substructural logics, from classical poly-modal logics to (bi-)intuitionistic modal logics to the Lambek calculus and its extensions, the Lambek-Grishin calculus, orthologic, the logic of (not necessarily distributive) De Morgan lattices, and the multiplicative-additive fragment of linear logic.
Alternation was first proposed by Chandra et al. for obtaining a theoretical model of parallel computations. Compared with nondeterminism, alternation gives computing devices the power of universal ...choice in addition to existential choice. In this paper, we put forward a notion of fuzzy alternating Büchi automata over a distributive lattice L (L-ABAs, for short). In our setting, a weight is the label of a leaf node of the run tree when executing a transition, which is helpful in complementing L-ABAs. Taking the dual operation on the transition function and negating final costs on states, we can get the complement of a given L-ABA. We point out that L-ABAs have the same expressive power as L-valued fuzzy nondeterministic Büchi automata (L-NBAs). A construction presented here shows that languages accepted by L-valued fuzzy alternating co-Büchi automata (L-ACAs) are also ω-regular languages. Moreover, closure properties of L-ABAs and decision problems for L-ABAs are also discussed.
•Put forward a notion of L-ABAs.•Show that L-ABAs have the same expressive power as L-NBAs.•Present closure properties of L-ABAs.•The languages accepted by L-ACAs are also ω-regular languages.•Discuss decision problems for L-ABAs.
Let L denote a finite lattice with at least two points and let A denote the incidence K-algebra of L over a field K. We prove that L is distributive if and only if A is an Auslander regular ring, ...which gives a homological characterisation of distributive lattices. In this case, A has an explicit minimal injective coresolution, whose i-th term is given by the elements of L covered by precisely i elements. We give a combinatorial formula of the Bass numbers of A. We apply our results to show that the order dimension of a distributive lattice L coincides with the global dimension of the incidence algebra of L. Also we categorify the rowmotion bijection for distributive lattices using higher Auslander-Reiten translates of the simple modules.
In this paper we introduce abstract string modules and give an explicit bijection between the submodule lattice of an abstract string module and the perfect matching lattice of the corresponding ...abstract snake graph. In particular, we make explicit the direct correspondence between a submodule of a string module and the perfect matching of the corresponding snake graph. For every string module we define a Coxeter element in a symmetric group. We then establish a bijection between the submodule lattice of the string module and the lattice given by the interval in the weak Bruhat order determined by the Coxeter element. Using the correspondence between string modules and snake graphs, we give a new concise formulation of snake graph calculus.
More on a curious nucleus Haykazyan, Levon
Journal of pure and applied algebra,
February 2020, 2020-02-00, Letnik:
224, Številka:
2
Journal Article
Recenzirano
Simmons (2010) 10 introduced a pre-nucleus and its associated nucleus that measure the subfitness of a frame. Here we continue the study of this pre-nucleus. We answer the questions posed by Simmons.
We treat the τ-tilting finiteness of minimal representation-infinite algebras and particularly the non-distributive ones. Building upon the new results of Bongartz, we fully determine which algebras ...in this family are τ-tilting finite and which ones are not. This complements our previous work in which we carried out a similar analysis for the minimal representation-infinite biserial algebras. Consequently, we obtain nontrivial explicit conditions for τ-tilting infiniteness of a large family of algebras. This also produces concrete families of “minimal τ-tilting infinite algebras”, recently studied in our work.
We further use our results to establish a conjectural connection between the τ-tilting theory and two geometric notions (the dense orbit property and Schur-representation finiteness) introduced by Chindris, Kinser and Weyman while studying module varieties. We verify the conjectures for the algebras studied in this note: For the minimal representation-infinite algebras which are non-distributive or biserial, if Λ has the dense orbit property, then Λ is τ-tilting finite. Moreover, we prove that such an algebra is Schur-representation-finite if and only if it is τ-tilting finite. This gives a categorical interpretation of Schur-representation-finiteness over this family of algebras.
The main goal of this paper is to give explicit descriptions of two maximal cones in the Gröbner fan of the Plücker ideal. These cones correspond to the monomial ideals given by semistandard and ...PBW-semistandard Young tableaux. For the first cone, as an intermediate result we obtain the description of a maximal cone in the Gröbner fan of any Hibi ideal. For the second, we generalize the notion of Hibi ideals by associating an ideal with every interpolating polytope. This is a family of polytopes that generalizes the order and chain polytopes of a poset (à la Fang–Fourier–Litza–Pegel). We then describe a maximal cone in the Gröbner fan of each of these ideals. We also establish some useful facts concerning PBW-semistandardness, in particular, we prove that it provides a new Hodge algebra structure on the Plücker algebra.
In this article we study the relations between three classes of lattices each extending the class of distributive lattices in a different way. In particular, we consider join-semidistributive, ...join-extremal and left-modular lattices, respectively. Our main motivation is a recent result by Thomas and Williams proving that every semidistributive, extremal lattice is left modular. We prove the converse of this on a slightly more general level. Our main result asserts that every join-semidistributive, left-modular lattice is join extremal. We also relate these properties to the topological notion of lexicographic shellability.
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