Semimodules over commutative semirings and modules over unitary commutative rings

We study the so-called closed and splitting subsemimodules and submodules of a given semimodule or module, respectively. We describe lattices of subsemimodules and of closed subsemimodules and posets of splitting subsemimodules and submodules. In the case of modules a natural bijective correspondence between these posets and posets of projections is established.


Introduction
It is well-known that any physical theory determines a class of event-state systems. To avoid details, in the case of quantum mechanics this event-state system is considered within the framework of a Hilbert space H whose projection operators are identified with the closed subspaces of H.
It was recognized in 1936 by Birkhoff and von Neumann [1] and 1937 by Husimi [2], see also [3] or [4], that if the Hilbert space H is of infinite dimension then the lattice of its closed subspaces need not be modular, contrary to the case of the lattice of all subspaces. However, a later inspection showed that a supremum need not exist when the subspaces are not orthogonal (cf. [5]. This was the reason why orthomodular posets were introduced (see e.g. [5]) and intensively studied during the last decades.
The natural question arises whether the property that the closed subspaces of H form an orthomodular lattice or an orthomodular poset is a privilege of a Hilbert space. It was already answered in the negative [6,7]: there are vector spaces that are not Hilbert spaces for which the splitting subspaces form orthomodular posets.
Since the tools for determining the orthomodular poset of splitting subspaces of a given vector space can be used also for modules and, more generally, for semimodules [8,9], we extend our study to closed subsemimodules and submodules. We define splitting subsemimodules and prove that for a given semimodule M, the set of its splitting subsemimodules forms a bounded poset with an antitone involution which, in the case when M is a module, turns out to be even an orthomodular poset. Similarly as for a Hilbert space, we use the method of projections and the bijective correspondence between the poset of projections and the poset of splitting submodules.
The concepts used for posets (i.e. ordered sets) and lattices are taken from monographs [5,10]. We hope that the study of closed and splitting subsemimodules and submodules and their lattices and posets can illuminate some properties of these concepts also in vector spaces, in particular in Hilbert spaces. Moreover, it may show that some physical theories need not be developed by using Hilbert spaces, but can be considered in a more general setting.

Semimodules over semirings
There are various definitions of a semiring in literature. We use that taken from the monograph [11].
Semimodules and semirings were studied by several authors, let us mention at least the papers [8,9,12,13]. Since these concepts are defined differently by the different authors, for the reader's convenience we provide the following definition. Definition 2.1: A semimodule over a commutative semiring (S, ⊕, ·, 0, 1) is an ordered quadruple (M, +, ·, 0) such that · is a mapping from S × M to M and the following conditions are satisfied for x, y ∈ M and a, b ∈ S: Recall that a subset U of a semimodule M = (M, +, ·, 0) over a commutative semiring (S, ⊕, ·, 0, 1) (or the corresponding ordered quadruple (U, +, ·, 0)) is called a subsemimodule of M if x + y, a x ∈ U for all x, y ∈ U and a ∈ S. Let L(M) denote the set of all subsemimodules of M.
Contrary to the case of vector spaces, not every semimodule has a basis. We define the notion of a basis for semimodules as follows.

Definition 2.2:
Let M = (M, +, ·, 0) be a semimodule over a commutative semiring (S, ⊕, ·, 0, 1) and I a non-empty set. Let In the following we will assume that M has a basis B. Then M is isomorphic to the subsemimodule (A, +, ·, 0) of (S, ⊕, ·, 0) I . Hence we may identify M with this subsemimodule. In the sequel we denote the coordinates of the element x of M with respect to the basis An example of a semimodule having a basis is the following. Let (S, ⊕, ·, 0, 1) be a commutative semiring and I = N. Then the subsemimodule (A, +, ·, 0) of (S, ⊕, ·, 0) N has the basis The situation is analogous for an arbitrary non-empty set I.
The concept of an inner product on semimodules was investigated in [9]. For the reader's convenience we recall the definition of the inner product as well as the concept of orthogonality for subsemimodules. We write x ⊥ y if x y = 0. Moreover, for C ⊆ M we put   We can describe the properties of the just defined concepts as follows.

Lemma 2.7:
Proof: (i) The first assertion is clear and the second easily follows by applying Proposition 2.5.
(ii) This follows from the fact that by Proposition 2.5, ⊥ is an antitone involution of (L c (M), ⊆).
Using Lemma 2.7 we obtain immediately The conditions in Theorem 2.9 are sufficient, but need not necessarily hold as the following example shows.
The Hasse diagram of (L(M), ⊆) is presented in Figure 1.
The lattice L(M) is not modular because it contains sublattices isomorphic to N 5 , e.g. the sublattice {U 1 , U 2 , U 4 , U 6 , M}. The unary operation ⊥ looks as follows: Figure 2.
It should be remarked that M does not satisfy the assumptions of Theorem 2.9 since  Hence M is the twodimensional vector space over the two-element field. Then M has the following subspaces: The Hasse diagram of (L(M), ⊆) is presented in Figure 3. The lattice L(M) is modular and the unary operation ⊥ looks as follows: It is easy to see that M satisfies the assumptions of Theorem 2.9.

Splitting subsemimodules
It can be easily checked that for a subsemimodule U of M, the semimodule U ⊥ need not be a complement of U in the lattice L(M) or L c (M), see e.g. Example 2.10. This is the motivation for introducing the following concept. Recall that if (P, ≤, 0, 1) is a bounded poset, then a unary operation on P is called a complementation if sup(x, x ) = 1 and inf(x, x ) = 0 for all x ∈ P. If is, moreover, an antitone involution then (P, ≤, , 0, 1) is called an orthoposet. In the sequel, we will denote sup and inf by ∨ and ∧, respectively, provided they exist. A mapping f from a poset (P, ≤) to a poset (Q, ≤) is called an antiisomorphism if it is bijective and if for all x, y ∈ P, x ≤ y is equivalent to f (y) ≤ f (x).
In the following theorem we use the formulation 'x i = 0 for almost all i ∈ I' which is the same as '{i ∈ I | x i = 0} is a cofinite subset of I'. It should be remarked that in any non-trivial bounded chain the smallest element is meet-irreducible.

The poset of projections
The next concept plays a crucial role in our study.   Moreover, we can prove the following.

Modules over rings
In this section we will investigate modules over unitary commutative rings instead of semimodules over commutative semirings. Of course, every module M over a unitary commutative ring S is a semimodule but now (M, +) is a commutative group. It means that on M there is also a binary operation − of subtraction. This enables us to reach stronger results than those above for semimodules.
In the sequel we assume that the semimodule M over the commutative semiring S is a module over the unitary commutative ring S, i.e. (M, +) is a commutative group.
In this section let L(M), L c (M) and L s (M) denote the set of all submodules, closed submodules and splitting submodules of M, respectively.
It is well known [14] that for a module M, the lattice L(M) is modular, unlike the case for semimodules (see Example 2.10). Proof: Let a, b ∈ M. Clearly, P is a linear mapping from M to itself, showing P ∈ Pr(M). Finally, Obviously, P U is a linear mapping from M to itself and (P U ) 2 = P U . Moreover, showing P U ∈ Pr(M). Now Corollary 5.7: If P, Q ∈ Pr(M) and P ⊥ Q then P ∧ Q = 0 and P ∨ Q = P + Q.
Recall from [5] that an orthomodular poset is a bounded poset (P, ≤, , 0, 1) with an antitone involution such that for all x, y ∈ P: if x ≤ y then x ∨ y exists, and if x ≤ y then y = x ∨ (y ∧ x ).
The notion of an orthomodular poset is well-defined: If x ≤ y then x ∨ y exists and hence x ∧ y exists, too. Moreover, x ∧ y ≤ x and hence (x ∧ y) ∨ x exists.
Our final result shows that the splitting submodules of M form an orthomodular poset. This was already shown for vector spaces over fields (cf. [6]).
The unary operation ⊥ looks as follows: Hence In our examples, the poset of splitting subsemimodules or splitting submodules is a lattice. In general, this need not hold. G. Birkhoff and J. von Neumann proved [1] that in the case of an infinite-dimensional Hilbert space over the field of complex numbers this poset is not a lattice but only an orthomodular poset. However, this need not hold only for Hilbert spaces. Posets of splitting subspaces which need not form lattices are intensively studied by Vetterlein (cf. [15]). However, up to now, we do not have an example of finite dimension.

Conclusion
We Open problems: (1) Find a module of finite dimension over a commutative ring whose poset of splitting submodules does not form a lattice. (2) Given an orthomodular poset P, find a module whose poset of splitting submodules is isomorphic to P.