Generalizing the concept of quantum triads

The concept of quantum triad has been introduced by D. Kruml [6], where for a given pair of quantale 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 , IV) 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 object A of C a left (right) module over M if there is a V-enriched functor from M (M) to C that maps ?M to A. Next, A is a M,N -bimodule if there is a V-enriched bifunctor from M ×N op to C mapping (?M , ?N ) to A. A map (in Set) f : |A|×|B| → |C| is a C-bimorphism[1] if every map f(a,−) : |B| → |C| and f(−, b) : |A| → |C| arises from a C-morphism fA : B → C or fB : A→ C, respectively, as |fA| or |fB |. Definition. 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 (S0, σ0) given by R⊗T L and σ0 : (r, l) 7→ r⊗T l, which is initial – for any solution (S, σ) there is a morphism s0 : S0 → S such that σ = s0 ◦ σ0. In this case, the solution belongs to C. Multiplication in S0 and action of S0 on R are as follows: • (r1 ⊗T l1) · (r2 ⊗T l2) = r1 · τ(l1, r2)⊗T l2 = r1 ⊗T τ(l1r2) · l2, • (r ⊗T l) · r′ = r · τ(l, r′) and l′ · (r ⊗T l) = τ(l, r′) · l. 2. S1 = {(α, β) ∈ C(L,L) × C(R,R) | τ(α(l), r) = τ(l, β(r)) for any l ∈ L, r ∈ R} with σ1 : (l, r) 7→ ((−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 S1. Multiplication and action of S1 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]. 208 N. Galatos, A. Kurz, C. Tsinakis (eds.), TACL 2013 (EPiC Series, vol. 25), pp. 208–210 Generalizing the concept of quantum triads Radek Šlesinger • (α1, β1) · (α2, β2) = (α2 ◦ α1, β1 ◦ β2), • (α, β)r = β(r) and l(α, β) = α(l). It has been shown in [6] that S0 and S1 form a so-called couple of quantales (see also [4]) and that S is a solution iff S0 → S1 factorizes through S. An example application in the original paper was reconstructing a quantale from given sets of its right-, leftand two-sided elements, and the base category was SupLat, complete joinsemilattices with join-preserving maps. Here we extend this construction to other categories. Example 1. Let V be a linear space over a field K, equipped with an inner product 〈−,−〉 : V × V → K. The solutions of the triad (V, V, 〈−,−〉,K) are then provided by linear operators on V , namely S0 consisting of elements u1 ⊗K v1 + · · ·+ un ⊗K vn, each acting on V as making a linear combination of u1, . . . , un with coefficients 〈vi, w〉, and S1 = {(α, β) ∈ End(V )×End(V ) | 〈α(u), v〉 = 〈u, β(v)〉}, the ring formed by pairs of adjoint linear operators on V (with reversed order of composition on the first component). In many situations, the categories C and V coincide, like in the original case of quantales (with SupLat, the category of complete join-semilattices), rings (with Ab, the category of abelian groups), or ordinary monoids (with Set), and the whole category of solutions is contained in C. Using more recent results on existence of tensor products for some algebras of quantum logic, other categories allowing formation of triads can be considered. For instance, we take the category effect algebras with effect algebra morphisms, EffAlg. Here the distinction of V and C is required as EffAlg has tensor products (when trivial effect algebras with 0 = 1 are allowed) and is monoidal with the 2-element algebra {0, 1} being the tensor unit, however, it is not closed [5]. The enriching category V is now that of generalized effect algebras (GenEffAlg). Example 2. A state (‘probability measure’) on an effect algebra E is an effect-algebra morphism (preserving ⊕ if it is defined, ⊥, 0 and 1) s : E → [0, 1]. A convex effect algebra is an effect algebra with action of the real interval [0, 1] (viewed as an effect algebra, with a ⊕ b defined if a + b ≤ 1), which becomes a monoid in EffAlg under standard multiplication of reals. Having two convex effect algebras with states (E, sE) and (F, sF ), one obtains an effect algebra with a state (E⊗F, sE · sF ) [2]. This forms a triad E×F → [0, 1]. Like in the previous example, its special solutions are S0 = F ⊗[0,1] E, and S1 – pairs of endomorphisms acting on the components of the tensor product, mutually adjoint wrt. the state on E ⊗ F .