In the present paper, the algorithmic correspondence theory developed in Conradie and Palmigiano [9] is extended to mu-calculi with a non-classical base. We focus in particular on the language of bi-intuitionistic modal mu-calculus. We enhance the algorithm ALBA introduced in Conradie and Palmigiano [9] so as to guarantee its success on the class of recursive mu-inequalities, which we introduce in this paper. Key to the soundness of this enhancement are the order-theoretic properties of the algebraic interpretation of the fixed point operators. We show that, when restricted to the Boolean setting, the recursive mu-inequalities coincide with the "Sahlqvist mu-formulas" defined in van Benthem, Bezhanishvili and Hodkinson [22]. IntroductionModal mu-calculus [17] is a logical framework combining simple modalities with fixed point operators, enriching the expressivity of modal logic so as to deal with infinite processes like recursion. It has a simple syntax, an easily given semantics, and is decidable. Modal mu-calculus has become a fundamental logical tool in theoretical computer science and has been extensively studied [3], and applied for instance in the context of temporal properties of systems, and of infinite properties of concurrent systems. Many expressive modal and temporal logics such as PDL, CTL, CTL * can be seen as fragments of the modal mu-calculus [3,15]. It provides a unifying framework connecting modal and temporal logics, automata theory and the theory of games, where fixed point constructions can be used to talk about the long term strategies of players, as discussed in [23].Correspondence theory studies the relationships between classical first-and second-order logic, and modal logic, interpreted on Kripke frames. A modal and a first-order formula correspond if they define the same class of structures. Specifically, Sahlqvist theory is concerned with the identification of syntactically specified classes of modal formulas which correspond to first-order formulas. Sahlqvist-style frame-correspondence theory for modal mu-calculus has recently been developed in [22]. Such analysis strengthens the general mathematical theory of the mu-calculus, facilitates the transfer of results ✩

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Sahlqvist-style correspondence results remain a perennial theme and an active topic of research within modal logic. Recently there has been interest in extending classical results in this area to the modal mu-calculus. We show how the `calculus of correspondence' and the ALBA algorithm (Conradie and Palmigiano, 2012) can be extended to the intuitionistic mu-calculus, and be used to derive FO+LFP frame correspondents for formulas of that logic. We define the class of recursive mu-inequalities, which we compare it with related classes in the literature including the Sahlqvist mu-formulas of van Benthem, Bezhanishvili and Hodkinson. We show that the ALBA algorithm succeeds in reducing every recursive mu-inequality, and hence that every recursive mu-inequality has a frame correspondent in FO+LFP.

Let K be a field. Let f ∈ K[[x 1 , ..., x r ]] and g ∈ K[[y 1 , ..., y s ]] be nonzero elements. If X (resp. Y) is a matrix factorization of f (resp. g), Yoshino had constructed a tensor product (of matrix factorizations) ⊗ such that X ⊗Y is a matrix factorization of f + g ∈ K[[x 1 , ..., x r , y 1 , ..., y s ]]. In this paper, we propose a bifunctorial operation ⊗ and its variant ⊗ ′ such that X ⊗Y and X ⊗ ′ Y are two different matrix factorizations of f g ∈ K[[x 1 , ..., x r , y 1 , ..., y s ]]. We call ⊗ the multiplicative tensor product of X and Y. Several properties of ⊗ are proved. Moreover, we find three functorial variants of Yoshino's tensor product ⊗. Then, ⊗ (or its variant) is used in conjunction with ⊗ (or any of its variants) to give an improved version of the standard algorithm for factoring polynomials using matrices on the class of summand-reducible polynomials defined in this paper. Our algorithm produces matrix factors whose size is at most one half the size one obtains using the standard method.

We use a matrix approach to study the concept of Morita context in the bicategory LG K of Landau-Ginzburg models on a particular class of objects. In fact, we first use properties of matrix factorizations to state and prove two necessary conditions to obtain a Morita context between two objects of LG K . Next, we use a celebrated result (due to Schur) on determinants of block matrices to show that these necessary conditions are not sufficient. Finally, we state a trivial sufficient condition.