We provide a generalization of Mundici's equivalence between unital Abelian lattice-ordered groups and MV-algebras: the category commutative is equivalent to MV-monoidal algebras. Roughly speaking, structures we call are without unary operation $x \mapsto -x$. The primitive operations $+$, $\lor$, $\land$, $0$, $1$, $-1$. A prime example these $\mathbb{R}$, with obvious interpretation operation...

Substructural logics extending the full Lambek calculus FL have largely benefited from a systematical algebraic approach based on the study of their algebraic counterparts: residuated lattices. Recently, a non-associative generalization of FL (which we call SL) has been studied by Galatos and Ono as the logics of lattice-ordered residuated unital groupoids. This paper is based on an alternative...

We introduce a relative semi-abelian category as a pair (C,E), where C is a pointed category with finite limits, and E is a class of normal epimorphisms in C satisfying certain conditions, stronger than those defining a relative homological category [2]. In the absolute case, i.e. when E is the class of all regular epimorphisms in C, the pair (C,E) is relative semi-abelian if and only if C is s...

Processes can be seen as relations extended in time. In this paper we want to investigate this observation by deriving an ordered category of processes. We model processes as co-algebras of a relator on Dedekind category up to bisimilarity. On those equivalence classes we define a lower semi-lattice structure and a monotone composition operation.

We define natural A∞-transformations and construct A∞-category of A∞-functors. The notion of non-strict units in an A∞-category is introduced. The 2-category of (unital) A∞-categories, functors and transformations is described. The study of higher homotopy associativity conditions for topological spaces began with Stasheff’s article [Sta63, I]. In a sequel to this paper [Sta63, II] Stasheff def...

We investigate the existence of normalizers of subobjects in pointed categories defined in the expected way, as motivated by the standard definition used in the category of groups. We show that, for a semi-abelian category C: (a) if the category C2 of morphisms in C is action representable, then so is the category Mon(C) of monomorphisms in C; (b) if Mon(C) is action representable, then normali...

In this paper we show that to a unital associative algebra object (resp. co-unital coassociative co-algebra object) of any abelian monoidal category (C,⊗) endowed with a symmetric 2-trace, i.e. an F ∈ Fun(C,Vec) satisfying some natural trace-like conditions, one can attach a cyclic (resp. cocyclic) module, and therefore speak of the (co)cyclic homology of the (co)algebra “with coefficients in F...

Consider decision problems derived from optimization problems. For problems in this category, we show that they cannot be expressed in existential second order (ESO) logic with a first order part that is universal Horn (or simply, ESO Π1 Horn) at the structure level, as opposed to machine level. This is true even when we consider ordered structures. We show that an NP-complete problem (Maximum ...