On a Characterization of the Lattice of Subsystems of a Transition System

It was first proved by Birkhoff and Frink, and the result now belongs to the folklore, that any algebraic lattice is up to isomorphism the lattice of subuniverses of a universal algebra. A study of subsystems of a transition system yields a new algebraic concept, that of a strongly algebraic lattice. We give here a representation theorem to the manner of Birkhoff and Frink of such lattices. A transition system is a pair (S, S), where (i) S is a set of states, (ii) S ⊆ S × S is the transition relation. We write s S s for (s,s) ∈ S. Nondeterministic transition systems, those (S, S) for which the set of successors of any element s ∈ S is an arbitrary set, are easily seen to be coalgebras of the covariant powerset functor ᏼ: Sets → Sets from the category of sets to itself. Observe that any unary algebra (S,Ᏺ) gives rise to a unique transition system (S, S), but the converse in the general case is false. A subsystem of a transition system (S, S) is a subset X of S which has the following stability property: s S s and s ∈ X imply s ∈ X. The empty set and the universe S are subsystems of (S, S), they are said to be trivial. It is straightforward to check that the set Subs(S) of subsystems of (S, S) is stable for arbitrary unions and intersections. Given a subset X of S, we denote by X the subsystem of (S, S) generated by X. It is the intersection of all subsystems of (S, S) containing X. The notation * S will be used to denote the reflexive and transitive closures of the binary relation

set Subs(S) of subsystems of (S, S ) is stable for arbitrary unions and intersections.
Given a subset X of S, we denote by X the subsystem of (S, S ) generated by X.It is the intersection of all subsystems of (S, S ) containing X.The notation * S will be used to denote the reflexive and transitive closures of the binary relation S on S. The subsystem X is then characterized as follows: Hence for s ∈ S, writing s the subsystem {s} , we get The mapping − : ᏼ(S) → ᏼ(S) defined from the set of subsets of S to itself is a closure operator on S. The previous characterization of X permits to see that We say that the closure operator − is completely additive.One can notice that (i) subsystems s of (S, S ), s ∈ S, satisfy the following finiteness condition: for all families (X i , i ∈ I) of subsystems of (S, S ) if s ⊆ i∈I X i , then there exists an index i 0 ∈ I such that s ⊆ X i0 , (ii) s ⊆ s if and only if s * S s .These observations prompt us to initiate the following definitions.
Definition 1.Let (E,≤) be an ordered set which admits arbitrary suprema.An element a in E is called s-compact (s for strongly compact), if for all covering a ≤ i∈I a i of a there exists an index i for which a ≤ a i .Consider a sup-complete lattice (E,≤) (i.e., an ordered set admitting arbitrary suprema).As a poset, (E,≤) can be viewed as a cocomplete category.Let a be in E, it is equivalent to say that a is s-compact or in categorical terms, every morphism f : a → colim I a i factors uniquely into a morphism f : a → a i (for some i ∈ I).This means that the covariant hom-functor [a,−] preserves all (small) colimits.Such an object a is called absolutely presentable (see [2]).Definition 2. A sup-complete lattice (L,≤) is called s-algebraic (or strongly algebraic), if each element a of L can be written as supremum of s-compact elements less than a.
Any s-algebraic lattice is obviously algebraic, but the converse is not true.In fact given a group (G, * ), the lattice (Sg(G),⊆) of subgroups of G is algebraic (see [1]).Further algebraic elements in (Sg(G)) are finitely generated subgroups of G.It is easy to verify that (Sg(Z,+),⊆) the lattice of subgroups of the additive group (Z,+) is not s-algebraic.
Consider the sup-complete lattice (L,≤) as a cocomplete category; it will be called salgebraic if every element in L is a colimit of absolutely presentable objects in L. Hence an s-algebraic lattice viewed as a category is locally absolutely presentable with the set of s-compact elements as set of absolutely presentable objects.
The basic example is that of a complete lattice of subsystems of a transition system; this seems also to be a generic s-algebraic lattice as shown by the following representation theorem.
Theorem 3. Let (L,≤) be an s-algebraic lattice.There exist a transition system (S, S ) and an isomorphism from L onto the lattice Subs(S) of subsystems of (S, S ).
Proof.We denote by S the set of s-compact elements of L. Define on S a binary relation S as follows: for all a,b ∈ S, a S b if and only if b ≤ a.Let ↓ x be the set of elements x ∈ L such that x ≤ x.For all x in L, the set S∩ ↓ x of s-compact elements less than x is a subsystem of S. In fact if a S b and a ∈ S∩ ↓ x, then we have b ≤ a, hence b ∈ S∩ ↓ x.
On deduces the mapping Let us check that ψ is order preserving and reflecting.To this end, let us consider x and x in L.
Conversely if ψ(x) ⊆ ψ(x ), since each element of L can be written as a supremum of s-compact elements less than itself, we have Finally let us show that ψ is a one-to-one mapping by exhibiting its inverse.For that set the mapping For all x ∈ L, we have φψ(x) = {a | a is s-compact and a ≤ x} = x.Further, for all subsystem X of (S, S ), It is clear that X ⊆ S ↓ X.Let a ∈ L such that a ≤ ∨X and a ∈ S. By s-compacity of a, there exists x ∈ X such that a ≤ x, that is, x S a by definition.Since X is a subsystem of (S, S ) and x ∈ X, we obtain a ∈ X.One deduces the inclusion S∩ ↓ X ⊆ X which induces the equality S∩ ↓ X = X, hence ψφ(X) = X.The fact that ψ preserves arbitrary suprema follows from the fact that each order isomorphism between complete lattices is automatically an isomorphism of complete lattice.The theorem is proved.
Since the s-algebraic lattice (L,≤) as a poset is a locally absolutely presentable category, it is isomorphic to the free cocompletion [S 0 ,Set] of the set S of s-compact elements, under all (small) colimits.This free cocompletion is, of course, isomorphic to the lattice of down-closed subsets of S which are precisely the subsystems of S. Therefore Theorem 3 gives a theoretical lattice version of the categorical well-known result stating that: every locally absolutely presentable category is isomorphic to the presheaf category.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.