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The symmetric monoidal closed category of cpo M-sets

In this paper, we show that the category of directed complete posets with bottom elements (cpos) endowed with an action of a monoid M on them forms a monoidal category. It is also proved that this category is symmetric closed.

cpos with an action of a monoid M on them. In other words, we study M -sets in the category Cpo. Among other things, we have shown that this category is not cartesian closed. In this paper, we show that the category Cpo Act-M is a symmetric monoidal closed category. In fact, it is wellknow that the category Cpo is a symmetric monoidal category. Hence and since Cpo Act-M ∼ = Cpo M op , it follows that the category Cpo Act-M is a symmetric monoidal category. Furthermore, we can not deduce the closedness of Cpo Act-M from the closedness of Cpo. Therefore, in the final section we prove that this category is closed. Because of the constructive proofs and descriptions through our manuscript, and the important role of domain theory in denotational semantics, we think that our results would be useful and interesting for theoretical computer scientists, as well as for algebraists and order theorists. More precisely, the action of a monoid on a set would always be an important concept for computer scientists, where they use this concept in automata theory. Moreover, the subject of finding a mathematical model for programming languages would also be an interesting and helpful tools for computer scientists. From this point of view, the domain theory was introduced as a mathematical model for semantics of programming languages [1]. Furthermore, there are many models for semantics of the programming language PCF (Programming Computable Functionals) which one of them is Domain models, that is, cpos with a family of actions of the natural numbers, where such cpos called SFP in the contexts, see [17]. By knowing SFP cpos, one can have, for example, any finite cpo N-act with the identity actions, as a SFP cpo, where N is considered with the binary operation min. On the other hands, an important problem in domain theory is the modelling of non-deterministic features of programming languages. There have been found some models in the literature, see, for example, [14][15][16]. In fact, to find such models, Plotkin and Smyth introduced the concept of a powerdomain, that is, a subset of a cpo, see [18]. In fact, they use the concept of a d-cone which is a commutative dcpo-monoid (in the sense of [11]) with an action of the monoid R + , of positive real numbers.
In the following we give some preliminaries needed in the sequel. Dcpos and cpos. First of all, we recall some basic concepts of posets, dcpos, and cpos. For more information one can see [1,4,7,8]. A partially ordered set (or a poset, for short) is a pair (A, ≤), where A is a set and ≤ is a binary relation on A which is also reflexive, antisymmetric, and transitive.
Let (A, ≤) be a poset and S ⊆ A. An element a ∈ A is said to be an upper bound of S if s ≤ a for each s ∈ S. Moreover, it is said to be the supremum or the join of S, denoted by S, if it is an upper bound of S and a ≤ b for each upper bound b of S.
A non-empty subset D of a partially ordered set (A, ≤) is called directed, denoted by D ⊆ d A, if for every a, b ∈ D there exists c ∈ D such that a, b ≤ c; and A is called directed complete, or briefly a dcpo, if for every D ⊆ d A, the supremum of D, denoted by d D (read the directed join of D), exists in A. A dcpo which has a bottom (least) element ⊥ is said to be a cpo.
A dcpo map or a continuous map f : A → B between dcpos is a map with the property that for every Thus we have the categories Dcpo and Cpo, of all dcpos and cpos with (strict) continuous maps between them, respectively.
The following lemmas are frequently used in this paper.
M -sets and cpo M -sets. Now, we recall the preliminary notions of the action of a monoid. For more information, see [6,9,10,12]. A monoid is a triple (M, * , 1), where M is a set, * is an associative binary operation on M , and 1 is an element of M called its identity element with the property that m * 1 = m = 1 * m, for all m ∈ M . From now on, whenever there is no confusion, we will write M for (M, * , 1) and also write m * n simply as mn. Also, by a cpo M -set map between cpo M -sets, we mean a strict continuous map which is also an M -set map. We denote the category of all cpo M -sets and cpo M -set maps between them by Cpo Act-M . Recall (see [10]) that this category is both complete and cocomplete.
Category Theory. Now, we recall from [2] the definitions of category and functor, for those who are not familiar with the subject. A category A consists of a class, also denoted by A, whose elements will be called objects of the category and for every pair A, B of objects, a set A(A, B), whose elements will be called morphisms or arrows from A to B, and also for every triple A, B, C of objects, there exists a composition law A(A, B) × A(B, C) → A(B, C), the composite of the pair (f, g) will be written g • f or just gf , which also satisfies the associativity axiom, that is, for arbitrary morphisms f ∈ A(A, B), g ∈ A(B, C), h ∈ A(C, D), the equality h•(g•f ) = (h • g) • f holds. Moreover, for every object A there exists a morphism Id A ∈ A(A, A), called the identity on A, which satisfies the usual identity axiom, that is, for every pair of morphisms f ∈ A(A, B), g ∈ A(B, C) the equalities Id B • f = f and g • Id B = g hold.
Also, a functor F from a category A to a category B consists of a mapping A → B between the classes of objects of A and B; the image of A ∈ A is written F A, and a mapping is written F f . Moreover, F must preserve the monoid structure on arrows, that is, for every pair of Monoidal closed category. Finally we recall the definition of a monoidal category from [3]. A monoidal category A is a category together with a bifunctor ⊗ : A × A → A, (A, B) → A ⊗ B, called the tensor product, an object I ∈ A, and three natural isomorphisms a ABC : satisfying the usual coherence axioms for a monoidal category (see axioms 4-5 in Definition 6.1.1 of [3]). If, furthermore, both -⊗ A and A ⊗ -have right adjoints for each A ∈ A, then A is called a biclosed category. A monoidal category (A, ⊗,

The category Cpo Act-M is a symmetric monoidal category
In this section, using the categorical properties of the category Cpo, we show that the category Cpo Act-M is a symmetric monoidal category.
First, we notice the following lemma with a sketch of its proof. Recall from [1] that the category Cpo is a symmetric monoidal category, in which the tensor product of two cpos A and B, which is also called smash product, is the cpo Therefore, applying Lemma 2.1 and Propositions 2.2, 2.3, we obtain the following theorem. 3 The closedness of the category Cpo Act-M In this section, we show that the symmetric category Cpo Act-M is closed. First, we recall from [3] and [2] the following two propositions.  In the following, unlike the above remark, we find a right adjoint for the functor A ⊗ -: Cpo Act-M → Cpo Act-M given in Theorem 2.4, for a general cpo M -set A. This proves that the symmetric monoidal category Cpo Act-M is closed. Proof. First notice that the (M ×(A\{⊥ A })) ⊥ with the pointwise order and action and the zero element ⊥ is a cpo M -set (see [10]). Now, we show that for a cpo M -set B, Hom where the third equality is true because B is a cpo M -set. Moreover,   Proof. We do this in four steps: Step (I). Defining counit: We show that for every cpo M -set B, the map where the forth equality is true because d F is calculated pointwise and each g ∈ F is continuous. Moreover, the fifth equality is true because x); and if g(1, x) ≤ b, for all (x, g) ∈ D and some b ∈ B, then d g∈F d x∈D g(1, x) ≤ b. To see this, let g ∈ F and x ∈ D , then there exist y ∈ A \ {⊥ A } and h ∈ Hom Cpo ((M × (A \ {⊥ A })) ⊥ , B)\{f ⊥ } such that (y, g) ∈ D and (x, h) ∈ D. Since D is directed, there exists (z, k) ∈ D with (y, g), (x, h) ≤ (z, k). This gives y, x ≤ z and g, h ≤ k. So g(1, x) ≤ g(1, z) ≤ k(1, z) ≤ b for all g ∈ F and x ∈ D , as required.
We also have d D = d D and then using the Case (1), Therefore, η B is a cpo M -set map.
Case (2) In the following, we see that, as one expects, for a cpo M -set A with the trivial actions, the two right adjoints Hom Cpo Act-M ((M × (A \ {⊥ A })) ⊥ , -) and (-) A to the functor A ⊗ -are isomorphic. Now, define the map