With this paper we hope to contribute to the theory of quantales and quantale-like structures. It considers the notion of Q-sup-algebra and shows a representation theorem for such structures generalizing the well-known representation theorems for quantales and sup-algebras. In addition, we present some important properties of the category of Q-sup-algebras.

The concept of quantum triad has been introduced by D. Kruml [6], where for a given pair of quantale 1 modules L, R over a common quantale Q, endowed with a bimorphism (a 'bilinear map') to Q, a construction equipping L and R with additional module structure and another bimorphism, both compatible with the existing bimorphism and action of the quantale, was presented. As the original concept was only defined in a specific setting of categories of quantale modules, we extend it to a more universal one, which can be applied to other common algebraic structures.In what follows, we assume that all the categories are concrete (via the forgetful functor | − | into Set). Let V = (V, ⊗ V , I V ) be a closed symmetric monoidal category and C be a subcategory of V, enriched in V. Suppose M is a monoid in V (viewed as a V-category with a single object M ). We call an objectDefinition. Let V, C be as above, L, R be a left and a right module over a V-monoid T . Further, let τ be a C-bimorphism from L × R to T . Then the tuple (L, R, τ, T ) is called a triad.If there exists a monoid S in V together with a V-bimorphism σ : R × L → S which makes L a T, S-bimodule and R a S, T -bimodule, and is compatible with τ (this means, for instance 'RLR': for any l ∈ L and r, r ∈ R, σ(r, l) · r = r · τ (l, r ), and a few similar conditions), we call (S, σ) a solution of the triad.Existence of solutions together with additional properties can be proved when certain assumptions on the category C are satisfied:Proposition. 1. Let C have tensor products over T , i.e., it has coequalizers of morphisms R ⊗ T ⊗ L ⇒ R ⊗ L obtained from T acting on R and L, respectively. Then the universal property of the tensor product provides a solution (S 0 , σ 0 ) given by R ⊗ T L and σ 0 : (r, l) → r ⊗ T l, which is initial -for any solution (S, σ) there is a morphism s 0 : S 0 → S such that σ = s 0 • σ 0 . In this case, the solution belongs to C. Multiplication in S 0 and action of S 0 on R are as follows:for any l ∈ L, r ∈ R} with σ 1 : (l, r) → ((−l)r, l(r−)) is another solution. It is terminal, since any monoid acting on L (R) and satisfying the compatibility conditions can be represented in S 1 . Multiplication and action of S 1 are following:1 A quantale is a complete join-semilattice equipped with associative binary multiplication distributing over arbitrary joins, or, a semigroup in the category of complete join-semilattices. For more information on quantales and their modules, see e. g. [7,8,9].

Abstract. If the standard concepts of partial-order relation and subset are fuzzified, taking valuation in a unital commutative quantale Q, corresponding concepts of joins and join-preserving mappings can be introduced. We present constructions of limits, colimits and Hom-objects in categories Q-Sup of Q-valued fuzzy joinsemilattices, showing the analogy to the ordinary category Sup of join-semilattices. IntroductionIn the standard concept of a fuzzy set [11], the relation "x is an element of X" is fuzzified, and replaced by a mapping X → [0, 1] assigning to each element of X its "membership degree". As further generalization, the membership degree can be evaluated in structures more general than the real unit interval, typically frames, residuated lattices, or quantales.In our paper, the concept of a set remains unchanged, and it will be the partial order relation and the notion of a subset that will be replaced by suitable mappings to a quantale. Instead of considering on X a partial order relation ≤, we employ a unital quantale Q and mappings M : X → Q and e : X × X → Q, which quantify the "degree of truth" of membership in a subset and of being less or equal.Sets equipped with quantale-valued binary mappings were initially investigated in the so-called quantitative domain theory [4]. A number of papers on the topic of sets with fuzzy order relations valuated in a complete lattice with additional structure have been published in recent years. Among many others, articles [9,10,12] may be used as a reference. There are various structures used for fuzzy valuation in the literature, e.g., frames [3] and complete residuated lattices [12], both being just special cases of quantales. Also terminology has not settled yet, and differs among authors. As the multiplicative unit of a quantale need not be its top element, even truth can have more degrees in Q. This is different from valuation using frames or residuated lattices where the unit is the top element as well.The basic properties of the category of complete join-semilattices as well as the fundamental constructions in this category such as limits and colimits have MSC (2010): primary 08A72, 06F99; secondary 18B35.