The Hahn embedding theorem for a class of residuated semigroups

Hahn's embedding theorem asserts that linearly ordered abelian groups embed in some lexicographic product of real groups. Hahn's theorem is generalized to a class of residuated semigroups in this paper, namely, to odd involutive commutative residuated chains which possess only finitely many idempotent elements. To this end, the partial lexicographic product construction is introduced to construct new odd involutive commutative residuated lattices from a pair of odd involutive commutative residuated lattices, and a representation theorem for odd involutive commutative residuated chains which possess only finitely many idempotent elements, by means of linearly ordered abelian groups and the partial lexicographic product construction is presented.


Introduction
Hahn's celebrated embedding theorem states that every linearly ordered abelian group G can be embedded as an ordered subgroup into the Hahn product where R is the additive group of real numbers (with its standard order), Ω is the set of archimedean equivalence classes of G (ordered naturally by the dominance relation), × Ω R is the set of all functions from Ω to R (alternatively the set of all vectors with real elements and with coordinates taken from Ω) which vanish outside a well-ordered set, endowed with a lexicographical order [12]. Briefly, every linearly ordered abelian group can be represented as a group of real-valued functions on a totally ordered set. By weakening the linearly ordered hypothesis, Conrad, Harvey, and Holland generalized Hahn's theorem for lattice-ordered abelian groups in [6] by showing that any abelian -group can be represented as a group of real-valued functions on a partially ordered set. By weakening the hypothesis on the existence of the inverse element but keeping the linear order, in this paper Hahn's theorem will be generalized to a class of residuated semigroups, namely to linearly ordered odd involutive commutative residuated lattices which possess only finitely many idempotent elements. The generalitation will be a by-product of a representation theorem for odd involutive commutative residuated chains which possess only finitely many idempotent elements, by means of the partial lexicographic product construction and using only linearly ordered abelian groups. Thus, the price for not having inverses in our semigroup framework is that our embedding is made into partial lexicographic products, introduced here, rather than lexicographic ones. The building blocks, which are linearly ordered abelian groups, remain the same.
Residuation is a basic concept in mathematics [3] with strong connections to Galois maps [11] and closure operators. Residuated semigroups have been introduced in the 1930s by Ward and Dilworth [22] to investigate ideal theory of commutative rings with unit. Recently the investigation of residuated lattices (that is, residuated monoids on lattices) has become quite intense, initiated by the discovery of the strong connection between residuated lattices and substructural logics [7] via an algebraic logic link [2]. Substructural logics encompass among many others, Classical logic, Intuitionistic logic, Lukasiewicz logic, Abelian logic, Relevance logics, Basic fuzzy logic, Monoidal t-norm logic, Full Lambek calculus, Linear logic, many-valued logics, mathematical fuzzy logics, along with their non-commutative versions. The theory of substructural logics has put all these logics, along with many others, under the same motivational and methodological umbrella, and residuated lattices themselves have been the key component in this remarkable unification. Examples of residuated lattices include Boolean algebras, Heyting algebras [16], MV-algebras [4], BL-algebras [13], and lattice-ordered groups, to name a few, a variety of other algebraic structures can be rendered as residuated lattices. Applications of substructural logics and residuated lattices span across proof theory, algebra, and computer science.
As for the structural description of classes of residuated lattices, non-integral residuated structures, and consequently substructural logics without the weakening rule, are far less understood at present than their integral counterparts. Therefore, some authors try to establish category equivalences between integral and non-integral residuated structures to gain a better understanding of the non-integral case, and in particular, to gain a better understanding of substructural logics without weakening [8,9]. Despite the extensive literature devoted to classes of residuated lattices, there are still very few results that effectively describe their structure, and many of these effective descriptions postulate, besides integrality, the naturally ordered condition 1 , too [1,4,10,13,15,17,18,20]. Linearly ordered odd involutive commutative residuated lattices (another terminology is odd involutive FL e -chains) are non-integral and not naturally ordered, unless they are trivial. Therefore, from a general point of view, our study is a contribution to the structural description of residuated structures which are neither naturally ordered nor integral.
involutive FL e -algebras x → x is somewhat reminiscent to a reflection operation across a point in geometry, yielding a symmetry of order two with a single fixed point. In this sense t = f means that the position of the two constants is symmetric in the lattice. Thus, an extreme situation is the integral case, when t is the top element of X and hence f is its bottom element. This case has been deeply studied in the literature. The other extreme situation, when "the two constants are in both the middle of the lattice" (that is, when t = f and so the algebra is odd) is a much less studied scenario. We remark that t < f can also hold in some involutive FL e -algebras. However, there is no third extremal situation, as it cannot be the case that f is the top element of X and t is its bottom element: For any residuated lattice with bottom element, the bottom element has to be the annihilator of the algebra, but t is its unit element, a contradiction unless the algebra is trivial.
Prominent examples of odd involutive FL e -algebras are odd Sugihara monoids and lattice-ordered abelian groups. The latter constitutes an algebraic semantics of Abelian Logic [19,5,21] while the former constitutes an algebraic semantics of IUML * , which is a logic at the intersection of relevance logic and many-valued logic [8]. These two examples represent two extreme situations from another viewpoint: There is a single idempotent element in any lattice-ordered abelian group, whereas all elements are idempotent in any odd Sugihara monoid. The scope of our investigation lies in between these two extremes; we shall assume that the number of idempotent elements of the odd involutive FL e -chain is finite. For this class a representation theorem along with some corollaries (e.g. the generalization of Hahn's theorem) will be presented in this paper.
2 Odd involutive FL e -algebras vs. partially ordered abelian groups We start with three preliminary observations, the first of which is being folklore (we present its proof, too, to keep the paper self contained).

(2.2)
If t ≤ f and the algebra is conic then y 1 > y implies Proof Using that is an involution one easily obtains (x * • y ) = (x * • y ) → * • f = x → * • (y → * • f ) = x → * • y. To prove (2.2) one proceeds as follows: 3) we proceed as follows. Since the algebra is involutive t = f holds. Since the algebra is conic, every element is comparable with f , too. Indeed, for any a ∈ X, if a were not compatible with f then since is an order reversing involution, a were not compatible with f = t, which is a contradiction. Hence, by residuation 4 , y 1 > y = (y ) = y → * • f implies In our investigations a crucial role will be played by the τ function.
Since t is the unit element of * •, Stab x is nonempty. Since * • is residuated, the greatest element of Stab x exists, and it holds true that 6. Since the order is linear, the opposite of the statement is τ ( Q.E.D. The notion of odd involutive FL e -algebras has been defined with respect to the general notion of residuated lattices by adding further postulates (such as commutativity, an extra constant f , involutivity, and the t = f property). The following theorem relates odd involutive FL e -algebras to (in the setting of residuated lattices, very specific) lattice-ordered abelian groups, thus picturing their precise interrelation. In addition, Theorem 2.4 will serve as the basic step of the induction in the proof of Theorem 11.1, too.
In the light of Theorem 2.4, in the sequel when we (loosely) speak about a subgroup of an odd involutive FL e -algebra, we always mean a cancellative subalgebra of it. Further, let Evidently, there is a subalgebra X gr of X over X gr 5 , and X gr is the largest subgroup of X. We call X gr the group part of X.

Two illustrative examples
Unfortunately, there does not exist any easily accessible example in the class of odd involutive FL e -chains, apart from the cancellative subclass (linearly ordered abelian groups, each with a single idempotent element) or the idempotent subclass (odd Sugihara monoids, all elements are idempotent). Here we start with the two simplest nontrivial examples of odd involutive FL e -chains over R, both of which have only a single strictly positive idempotent element. These examples will be the inspirational source for the type I-IV partial lexicographic product constructions in Definition 4.2.
Using (2.2) it takes a half-line proof to show that for any odd involutive FL ealgebra, the residual complement of any negative idempotent element u is also The author of the present paper has put a considerable effort into proving the converse of this statement, with no avail. In fact, the converse statement does not hold, as shown by the following counterexample.
Example 3.1 The odd involutive FL e -chain which is to be defined in this example is Z Z Z ← × R R R (up to isomorphism), see item B in Definition 4.2 for its formal definition. Here we present an informal account on the motivation and considerations which had led to the discovery of this example.
Let I = ]-1, 0[. Consider any order-isomorphic copy I = (I, , -1 2 ) of the linearly ordered abelian group R R R = (R, +, 0) 6 . Our plan is to put a series of Figure 1: Visualization: The graph of the operation copies of I in a row (after each other, one for each integer, that is, formally we consider Z × I) to get R, and to define, based on , an operation * • over R so that the resulting structure becomes an odd involutive FL e -chain. The first problem is caused by the fact that whichever involutive structure we start with (in place of I), the above-described repetition of its universe will never yield R. In our example I = ]-1, 0[ does not have top and bottom elements, hence by the above-described repetition of it we obtain a set which is order-isomorphic to R \ Z, and not to R. Or if the original structure does have a top element (and hence by involutivity a bottom element, too) then the universe resulted by the above-described repetition will have gaps. To overcome this we extend to a new universe I := I ∪ {0} = ]-1, 0] by letting 0 be an annihilator of . In other words, we add a new top element to the structure I. The resulting structure, denote it by I , will no longer be an odd involutive FL e -chain, but it is still an FL e -chain, albeit obviously not even involutive due to the broken symmetry of its underlying set. Luckily we can immediately eliminate this broken symmetry by putting ℵ 0 copies of I =]-1, 0] in a row, thus obtaining R. Now we can define an operation * • over R by letting for a, b ∈ R, (see the graph of an order-isomorphic copy of * • in Figure 2, right, for the use of such 3D plots the interested reader is referred to [14]). The slightly cumbersome task of verifying that (R, ≤, * •, → * • , -1 2 , -1 2 ) is an odd involutive FL e -chain, is left to the reader. It amounts to verify the associativity of * • 8 , to compute the residual operation of * •, to confirm that the residual complement operation is x = -x − 1, and to confirm that -1 2 is the unit element and also the fixed point of . Note that the only strictly positive idempotent element of the constructed algebra is 0, while -1 2 is its unit element. Note, however, that the residual complement of 0 is not idempotent: 0 = -1 and -1 * • -1 = -2.
In the quest for a deeper understanding of this example, observe that apart from , the addition operation of R R R plays a role in the definition of * •. Moreover, by using a particular order-isomorphism, we can define * • using and the addition operation of Z Z Z only, as follows. As said, putting a series of copies of I in a row can, formally, be obtained by considering the lexicographic product Z ← × I . Then, h(a) = ( a , a − a ) is an order-isomorphism from R to Z ← × I . It is left to the reader to verify that * • can equivalently be defined in a coordinatewise manner by letting a * where h(a) = (a 1 , a 2 ) and h(b) = (b 1 , b 2 ). By using the natural order-reversing involutions (a * = -a on Z and x = -x − 1 on I) the reader can also verify that the residual operation of * • is given by if y ∈ I . (3.1) With this notation, the only strictly positive idempotent element of the constructed algebra is (0, 0), while (0, -1 2 ) is its unit element. A moral of this example is that the latter coordinatewise formalism for * • enlightens the example by providing a deeper insight into and a clear description of why * • makes up for an odd involutive FL e -chain (why is it associative, residuated, etc.) Compare , thus the associativity of ensures the associativity of * •. 9 Compare (3.1) with the last two rows of (4.2); for the last row of (4.2) see Definition 4.1 for the notation ↓ .
necessarily have to work with groups (like Z Z Z and R R R in our example), but also odd involutive FL e -algebras suffice in both coordinates. But in this case only the group part of the first algebra (and no more) can be enlarged. This makes our lexicographic product construction partial in nature, as if the first algebra is not a group then its group part is strictly smaller than the whole algebra. Leaving the rest of the first algebra unchanged will, formally, mean that we consider a Cartesian product there, too, to enable a coordinatewise treatment of the operations, namely, the Cartesian product of the rest of the first algebra with a singleton. This example has inspired the type II partial lexicographic product construction.
Moreover, it turns out that instead of enlarging the group part one can enlarge any subgroup of the group part. This makes our lexicographic product construction definitively partial in nature, since even if the first algebra is a group (and hence its group part is the whole algebra), an even smaller algebra (a subalgebra of the group part) can be enlarged, by the second algebra extended by a top. This option of partial enlargement inspires the type IV partial lexicographic construction defined in item (B) of Definition 4.2.
Example 3.2 The odd involutive FL e -chain which is to be defined in this ex- In order to further exploit the partial nature of our lexicographic product construction, take the subgroup Z Z Z = (Z, +, 0) of the linearly ordered abelian group of the reals R R R = (R, +, 0). Our plan is to replace each integer inside the reals by a whole copy of the reals (isomorphically, by I from I = (I, , -1 2 ) of Example 3.1), and to define an odd involutive FL e -algebra over it based on . Again, if we wish to obtain a universe, which is order isomorphic to R, then replacing each integer inside the reals by I = ]-1, 0[ is not convenient; the resulting universe will not be topologically connected, as it is easy to see. To overcome this we extend to I ⊥ := I ∪ {-1, 0} = [-1, 0] by letting first 0 be annihilator over ]-1, 0], and then -1 be annihilator over [-1, 0]. In other words, we add a new top and a new bottom to the structure I, thus obtaining I ⊥ . Now, let the new universe be Z × I ⊥ ∪ (R \ Z) × {-1}. Again, we define * • coordinatewise by letting (a 1 , a 2 ) * • (b 1 , b 2 ) = (a 1 + b 1 , a 2 b 2 ), and we define , using the natural order-reversing involutions (a * = -a on R and x = -x − 1 on I ⊥ ) also coordinatewise by What we obtain is an odd involutive FL e -chain, see the graph of an orderisomorphic copy of * • in Figure 2, left. Here, too, there is only one strictly positive idempotent element in the constructed algebra, namely (0, 0), while (0, -1 2 ) is its unit element. Note that unlike in the previous example here the residual complement of (0, 0) is idempotent: (0, 0) = (0, -1) and (0, -1) * • (0, -1) = (0, -1).
Compare the example above with the formal definition of R R R Z Z Z ← ← × R R R in item A in Definition 4.2. Further inspection reveals that in these kind of examples one does not necessarily have to work with groups (like R R R together with its subgroup Z Z Z in the first coordinate, and R R R in the second), but any odd involutive FL ealgebra with any of its subgroup suffices in the first coordinate, and any odd involutive FL e -algebra suffices in the second one. Summing up, we start with an odd involutive FL e -algebra and we only enlarge a subgroup of it by an arbitrary odd involutive FL e -algebra equipped with top and bottom. This example inspired the type I partial lexicographic product construction.
Moreover, it turns out that instead of enlarging a subgroup, one can (1) enlarge only a subgroup of a subgroup by the second algebra equipped with top and bottom, plus (2) enlarge the difference of the two subgroups by only the top and the bottom. This second step is equivalent to enlarging the difference of the two subgroups also by the second algebra equipped with top and bottom, and then removing the second algebra, thus leaving only its top and bottom there. Ultimately, at this removal step we take a subalgebra compared to the corresponding algebra which constructed by the type I variant, see Theorem 4.4. The option of partial enlargement (when only a subalgebra of a subalgebra is enlarged) in this example inspires the type III partial lexicographic construction defined in item (A) of Definition 4.2.
4 Constructing involutive FL e -algebras -Partial lexicographic products We start with a notation. x ↓ = z if there exists a unique z ∈ X such that x covers z, x otherwise.
We define x ↑ dually. Note that if is an order-reversing involution of X then it holds true that Next, we introduce the construction for the key result of the paper, called partial lexicographic product (or partial lexicographic extension) with four slightly different variations in Definition 4.2. Roughly, only a subalgebra is used as a first component of a lexicographic product and the rest of the algebra is left unchanged, hence the adjective 'partial'. This results in an involutive FL e -algebra, which is odd, too, provided that the second component of the lexicographic product is so, see Theorem 4.4.
In Remark 4.5 we shall refer to some algebraic notions which appear later in the paper at the algebraic decomposition part, thus trying to make a bridge between the different components of the construction part and the decomposition part of the paper.
× Y, the type III partial lexicographic product of X, Z, V and Y as follows: In the particular case when V = Z, we use the simpler notation × Y and call it the type I partial lexicographic product of X, Z, and Y.
B. Assume that X gr is discretely embedded into X. Add a new element to Y as a top element and annihilator. Let V ≤ X gr . Let and define X V ← × Y, the type IV partial lexicographic product of X, V and Y as follows: where ≤ is the restriction of the lexicographical order of ≤ X and ≤ Y ∪{ } to X V ← × Y , * • is defined coordinatewise, and the operation → * • is given by In the particular case when V = X gr , we use the simpler notation X ← × Y for X V ← × Y and call it the type II partial lexicographic product of X and Y.
In order to reduce the number of different cases to be considered when checking the residuated nature of some structure (e.g. in the proof of Theorem 4.4) we will rely on the following statement, which generalizes the well-known an equivalent formulation of involutive FL e -algebras in terms of dualizing constants.
If there exists a dualizing element c of M then * • is residuated, and its residual operation is given by 10 Note that intuitively it would make up for a coordinatewise definition, too, in the second line of (4.2) to define it as (x * , ⊥). But ⊥ is not amongst the set of possible second coordinates. However, since Xgr is discretely embedded into X, if (x * , ⊥) would be an element of the algebra then it would be equal to ((x * ) ↓ , ). 11 Dualizing elements are defined in residuated structures in the literature, see e.g. [7, Section 3.4.17.]. 12 Here we do not assume that * • is residuated. We only postulate that the greatest element of the set {z ∈ M | x * • z ≤ c} exists for all x and a fixed c. Q.E.D.
Theorem 4.4 Adapt the notation of Definition 4.2.
the only non-trivial cases are (i) when the product of the first coordinates in not in Z, because then the product of the second coordinates has to be ⊥, and (ii) when the product of the first coordinates in in Z \V , because then the product of the second coordinates has to be or ⊥. But if the product of the second coordinates is not ⊥ then none of the second coordinates can be ⊥ (since ⊥ in annihilator), hence both first coordinates must be in Z, as required. Second, if the product of the second coordinates is neither nor ⊥ then none of the second coordinates can be or ⊥ (because of the product table for and ⊥), hence both first coordinates must be in V , as required.
Since ⊥ is annihilator, * • can be expressed as hence associativity of * • and that (t X , t Y ) is the unit of * • are straightforward.
Next, we state that * • is residuated. By Proposition 4.3 it suffices to prove -Assume that at least one of x 1 and x 2 is not in Z. Due to commutativity of * and , and the involutivity of , we may safely assume x 2 ∈ Z. Then y 2 = ⊥ and thus ( , . . .) = (x 1 , y 1 ) , and we are done.
This holds if and only if either This holds if and only if either It is straightforward to verify that is an order-reversing involution on X V ← × Y at the first and third rows of (4.2). At the second row of (4.2), we can use that X gr is discretely embedded into X and so (4.1) holds for x ∈ X gr .
Since annihilates all elements of Y , * • can be expressed as hence associativity of * • and that ((t X , t Y )) is the unit of * • are straightforward.
Next, we state that * • is residuated. By the claim, it suffices to prove that , y 1 ) = (x 1 , y 1 ) and we are done. The case y 1 = and y 2 ∈ Y is completely analogous.
-Finally, assume y 1 , y 2 = . Then, since > f Y , it holds true that The case x 2 ∈ X gr is completely analogous. -If Since type I and II partial lexicographic products are particular cases of type III and IV partial lexicographic products, it follows from the previous two claims that From the respective definition of the residual complement operations it also easily follows that In the rest of the paper we will not use the partial lexicographic product construction in its full generality. For describing the structure of odd involutive FL e -chains with only finitely-many idempotent elements, first of all, we will apply it to the linearly ordered setting only. Second, it will be sufficient to use FL e -chains with finitely many positive idempotent elements and linearly ordered abelian groups to play the role of X and Y, the algebras in the first and in the second coordinate, respectively.
Keeping it in mind, in order to ease the understanding of the rest of the paper, we make the following Remark 4.5 Concerning Definition 4.2: for v ∈ V , will be recovered in an arbitrary odd involutive FL e -chain X in Section 6, and will be called convex components of the group part of X.
• The inverse operation of the enlargement of each element of V by Y will be done in Section 6 by a homomorphism β which collapses each convex component of the group part of X into a singleton.
• In case the residual complement of the smallest strictly positive idempotent element of X is also idempotent then the homomorphism γ • β will collapse each convex component of the group part of X together with its infimum and supremum (in other words, together with its extremals, see below) into a singleton in Section 9.
• Those elements of and those elements of X V ← × Y which are of the form (v, ) for v ∈ V will be recovered in an arbitrary odd involutive FL e -chain in Section 7, and will be called extremals.
• Those elements of X Z V ← ← × Y which are of the form (z, ) or (z, ⊥) for z ∈ Z \ V , and those elements of X V ← × Y which are of the form (z, ) for z ∈ X gr \ V will be recovered in an arbitrary odd involutive FL echain in Section 8, and will be called pseudo extremals. We will call these elements pseudo extremals, since there is no convex component in the algebra × Y for which they would serve as infimum or supremum (i.e., extremal). Their convex component could be there (if it were we would talk about a type I or a type II extension) but it is 'missing' (and thus we have a type III or a type IV extension, only a subalgebra of the related type I or a type II extension, see the last statement in Theorem 4.4).

Subuniverses
Sketch of the main theorem (Theorem 11.1) and its proof. For any odd involutive FL e -chain X, such that there exists its smallest strictly positive idempotent element, we introduce two different decomposition methods in Lemmas 9.4 and 10.2. The algebra will be decomposed by either the first or the second variant depending on whether the residual complement of its smallest strictly positive idempotent element is idempotent or not.
If the residual complement of the smallest strictly positive idempotent element is idempotent then 1. we will define two consecutive homomorphisms β and γ, and will isolate a homomorphic image γ(β(X)) of X, which has one less positive idempotent elements then X. In this sense γ(β(X)) will be smaller than the original algebra X.
If the residual complement of the smallest strictly positive idempotent element is not idempotent then 1. we will isolate a residuated subsemigroup X τ ≥u of X, which will be an odd involutive FL e -algebra, albeit not a subalgebra of X, since its unit element and residual complement operation will differ from those of X. X τ ≥u has one less positive idempotent element then X, so here too, X τ ≥u will be smaller than the original algebra X.
2. Then we prove that the group-part of X τ ≥u is discretely embedded into X τ ≥u , and we isolate a subgroup X Tc τ ≥u of it. 3. Finally we will recover X, up to isomorphism, as the type IV partial lexicographic product of X τ ≥u , X Tc τ ≥u and ker β in Section 10. Therefore, if the original odd involutive FL e -chain X has finitely many positive idempotents then (since then there exists its smallest strictly positive idempotent) we can reconstruct X from a smaller odd involutive FL e -chain via enlarging it by a linearly ordered abelian group (either type III or IV in Sections 9 and 10, respectively). By applying this decomposition iteratively to the obtained smaller algebra we end up with an odd involutive FL e -chain with a single idempotent element, which is a linearly ordered abelian group by Theorem 2.4. Summing up, any odd involutive FL e -chain which has only finitely many positive idempotent elements will be described by the consecutive application of the partial lexicographic product construction using only linearly ordered abelian groups as building blocks; as many of them as the number of the positive idempotent elements of X.
To this end first we need the classification of elements of X as described in Definition 5.1.
It is well-known that for any involutive FL e -chain, if c and c are idempotent and the interval between them contains the two constants then there is a subalgebra over that interval. However, in order to describe the structure of odd involutive FL e -chains these subalgebras will not be convenient to consider, they are too small. Rather, it will be wiser to put all elements of X sharing the same τ -value into one class: For an odd involutive FL e -chain (X, ≤, * •, → * • , t, f ), for u ≥ t and ∈ {<, =, ≥} denote The definition above reveals the true nature of τ : note that τ acts as a kind of 'local unit' function: τ (z) is the unit element for the subset X τ ≥τ (z) of X.
2. X τ =t = X gr and hence X τ =t is a nonempty subuniverse of a linearly ordered abelian group. For x ∈ X τ =t , the map y → y * • x (y ∈ X) is strictly increasing.
The respective subalgebras of X will be denoted by X τ <u and X τ =t = X gr . In the proof of Proposition 5.2 and also later a key role will be played by Lemma 5.4. To prove it, it will be useful to observe first Proof By (2.2), (x * • y ) ≥ x * •y holds, hence it suffices to prove x 1 * •y 1 > (x * • y ) . Assume the opposite, which is x 1 * •y 1 ≤ (x * • y )→ * • f since (X, ≤) is a chain. By adjointness from this we obtain (x * • x 1 ) * •(y * • y 1 ) = ( On the other hand, from Claims (1) and (3) in Proposition 2.3 show how τ behaves in relation to and * •, respectively. If the algebra is odd we can state more. Proof First we claim τ (x * • y) = max(τ (x), τ (y)) for x, y ∈ X. By claim (3) in Proposition 2.3, τ (x * • y) ≥ max(τ (x), τ (y)) holds. Assume τ (x * • y) > max(τ (x), τ (y)) and let z ∈] max(τ (x), τ (y)), τ (x * • y)]( = ∅) be arbitrary. By (2.4) it holds true that t ≤ τ (x) < z, hence the isotonicity of * • it holds true that x = t * • x ≤ z * • x and y = t * • y ≤ z * • y. Since τ assigns to x the greatest element of the stabilizer set of x, and z > max(τ (x), τ (y)), it follows that z does not stabilize x or y. Consequently, x < z * • x and y < z * • y should hold. Since t < z and z ≤ τ (x * • y) yields z ∈ Stab x * •y , it follows that  (1) Hence τ (x) = t, and thus x ∈ X τ =t too. These prove X τ =t = X gr . The rest of claim 2 follows from the cancellation property in groups. Q.E.D.
6 Collapsing the convex components of the group part of X into singletons -the homomorphism β Roughly, in algebra under a natural homomorphisms each element is mapped into its class, and the classes are often of the same size. In Definition 6.1 it is not the case: some elements will be mapped into their class (into their respective convex component with respect to τ < u, see below), while on other elements the map will be the identity map. In order to keep notation as simple as possible, these elements will be mapped into sets, too, namely to a singleton containing the element itself.
Definition 6.1 Let (X, ≤, * •, → * • , t, f ) be an odd involutive FL e -chain with residual complement , u > t idempotent. For x, y ∈ X τ <u , define x ∼ y if z ∈ X τ <u holds for any z ∈ X with x < z < y. It is an equivalence relation on X τ <u since the order is linear. Denote the component of x by [x] τ <u and call it the convex component of x with respect to τ < u. If u is clear from the context we shall simply write [x].
For z ∈ X let For Z ⊆ X let β(Z) = {β(z) : z ∈ Z}. For x, y ∈ β(X) let p ∈ x, q ∈ y (equivalently, let p, q ∈ X, x = β(p) and y = β(q)), and over β(X) define x ≤ β y iff p ≤ q, (6.1) x * • β y = β(p * • q), (6.2) 3) Although the constructions will be different for the two cases (depending on the idempotency of the residual complement of the smallest strictly positive idempotent element in X), in both cases the proof relies on Lemma 6.2. The map β which makes each convex component of X gr collapse into a singleton (its convex component with respect to τ < u), and leaves the rest of the algebra unchanged, will be shown to be a homomorphism.

3.
[t] τ <u = ]u , u[ and it is a nonempty subuniverse of X; denote the related subalgebra by ker β .
4. If u is the smallest strictly positive idempotent element then X τ <u = X gr , ker β is a subgroup of X gr , and (qua linearly ordered abelian groups) X gr β(X gr ) ← × ker β .
(2) In the light of the previous claim, to verify that β(X gr ) is a subalgebra of β(X) it suffices to note that X gr is a nonempty subuniverse of X by Proposition 5.2. By Theorem 2.4, showing cancellativity of β(X gr ) is equivalent to showing that each of its elements has inverse; but it is straightforward since each element of X gr has inverse and a homomorphism preserves this property.
(  (1) and (6) in Proposition 2.3. On the other hand, since u is idempotent, by claims (1) and (4)  (4) For x ∈ X τ <u , τ (x) > t would yield τ (x) be an idempotent by claim (4) in Proposition 2.3, which is strictly between t and u, contrary to assumption. Hence X τ <u = X τ =t holds. Claim (2) in Proposition 5.2 concludes the proof of X τ <u = X gr . Therefore, X τ <u is the group part of X. By claim (3), [t] τ <u is a subuniverse of X, hence it follows from X τ <u = X gr that [t] [τ <u] is a universe of a linearly ordered group, and thus ker β is a convex subgroup of X τ <u .
By the first isomorphism theorem, (qua linearly ordered abelian groups) β(X gr ) X gr /ker β . By Hahn's theorem, X gr and ker β are isomorphic to Hahn products of real groups. In particular, let X gr ← H × i∈I X i . Since convex subgroups are unions of whole archimedean classes, and since ker β is a convex subgroup of X gr , it follows that ker β is a union of whole archimedean classes of X gr . Therefore, since in Hahn products elements of the index set correspond to archimedean classes, it follows that there exists a subset J of I such that ker β ← H × i∈IXi , whereX i = X i if i ∈ J andX i is trivial if i ∈ I \J (in case of just assumingX i ≤ X i for i ∈ J, the archimedean classes would not be whole). Since ker β is convex, J must be a downset of I. Therefore, ker β ← H × i∈J X i , and hence X gr /ker β Q.E.D.

Extremals
In this section we define and investigate the extremal elements of the convex components of the group part of X. Definition 7.1 Let (X, ≤, * •, → * • , t, f ) be an odd involutive FL e -chain and u > t be idempotent. Provisionally, for v ∈ X τ <u let Call [v] and ⊥ [v] the top-and the bottom-extremals of the (convex) component of v. Denote The τ ≥ u subscript in the notations refers to the fact that X Tc τ ≥u , X Bc τ ≥u , and X Ec τ ≥u are subsets of X τ ≥u . We shall shortly prove it in Proposition 7.2 Let X = (X, ≤, * •, → * • , t, f ) be an odd involutive FL e -chain such that there exists u, the smallest strictly positive idempotent element, and let v ∈ X τ <u .
It holds true that 3. The following product table holds true for * •: If v, w ∈ X τ <u , a ↓ = a ∈ X τ ≥u \ X Tc τ ≥u , and y ∈ X τ ≥u then the following holds true. [v]

Proof
(1): By claim (3) Since for z ∈ [t], the mapping z → v * • z is a bijection between the two cosets [t] and [v], it follows that [v] has the same cardinality as [t].
z, thus confirming the existence of [v] along with (7.1) in the case of u = t ↑ . Next we prove, the existence of ⊥ [v] if u = t ↑ . Because of (6.4), . Therefore z ∈[v] z exists and since is an order reversing involution of X, it follows that ( z ∈[v] z) is equal to . The previous argument also shows (7.3) if u = t ↑ . (7.3) readily follows from (4.1) if u = t ↑ . Next we prove (7.1) and (7.2) when u = t ↑ . By monotonicity of * It remains to prove (7.2) in the u = t ↑ case. By (7.3), (7.1), and (2.2) ) . It follows that there is equality everywhere, hence (7.2) follows.
(2): By (7.1), Lemma 5.4, and claim (4) (3): Grey items in Table 1 are either straightforward or follow from the black items thereof by commutativity. Therefore, we prove only the black items.
where the latest equality is implied by y ∈ X τ ≥u ; this proves Table 1 (3,4) .
-There exists Z 2 ⊂ X \ {a } such that a = Z 2 , since a ↑ = a . We may safely assume Z 2 ⊂ X τ ≥u \ {a } since a is not the greatest lower bound of any component of X τ <u . Therefore, by (2.3), Table 1 (3,4) , and (2.2), respectively, for z ∈ Z 2 it holds true that a *  (2,3) and Table 1 (2,1) , for any v ∈ X τ <u , hold, thus the proof of the claim is concluded. Therefore, it suffices to prove that would follow, a contradiction to the idempotency of ⊥ [t] . Q.E.D.

Gaps -Motto: "Not all gaps created equal"
In order to treat the case when the order density of X is not assumed (c.f. Table 1 (4,1) , where a kind of density condition must have been assumed for a), first we classify gaps in X τ ≥u . It turns out that some gaps behave like extremal elements of "missing" components of X, those will be called pseudo extremals.
Definition 8.1 Let X = (X, ≤, * •, → * • , t, f ) be an odd involutive FL e -chain with residual complement , such that there exists u, the smallest strictly positive idempotent. Let For x ∈ X τ ≥u denote x ⇓ the predecessor of x inside X τ ≥u , that is let z if there exists X τ ≥u z < x such that there is no element in X τ ≥u strictly between z and x, x if for any X τ ≥u z < x there exists v ∈ X τ ≥u such that z < v < x holds. Clearly, Define x ⇑ dually.
In the third case, either x ∈ X Tps τ ≥u or x ∈ X G2 τ ≥u holds by item (2). Hence, since X G2 τ ≥u ∩ X T τ ≥u = ∅ by claim (1), to prove (8.1) it remains to prove The proof of (8.1) is concluded.
(7): Grey items in Table 2 are either straightforward, or can readily be seen by the commutativity of * •, or are inherited from Table 1, so it suffices to prove the items in black.
Q.E.D.  If v, w ∈ X τ <u , x, y ∈ X Tps τ ≥u , z, s ∈ X τ ≥u \ X T τ ≥u then the following holds true.
Proof Grey items in Table 3 are either inherited from Table 2, or straightforward, or are readily seen by the commutativity of * •. Hence it suffices to prove the items in black.
• Finally, we prove the remaining 2×2 entries at the bottom-right of Table 3.
The set of positive idempotent elements of Y is order isomorphic to the set of positive idempotent elements of X deprived of t.

Over
Proof 1. Because of claim (1) in Lemma 6.2, it suffices to verify that γ is a homomorphism from β(X) to γ(β(X)). The definition of ≤ γ is independent of the choice of x and y, since the sets and {{p}, {p ↓ }} (for p ∈ X Tps τ ≥u ) are convex in β(X). The ordering ≤ β is linear since so is ≤. Therefore, (9.2) is well-defined. Preservation of the monoidal operation can readily be checked in Table 3, and preservation of the residual complement can readily be seen from (6.4), (7.3) and claim (4) in Proposition 8.2. In complete analogy to the way we proved the preservation of the implication under β in (6.5), we can verify the preservation of the implication under γ.
If c ≥ u is an idempotent element of X then γ(β(c)) is also idempotent and γ(β(c)) ≥ γ(β(u)) = γ(β(t)) (thus also positive) since γ • β is a homomorphism. Assume c, d ≥ u are idempotent elements of X and c < d. Then c, d ∈ X τ ≥u holds because of claim (4) in Proposition 2.3. Therefore, referring to the definition of β and γ, and claim (2) in Proposition 7.2, γ(β(c)) = γ(β(d)) could hold only if either c = ⊥ [v] and d = [v] for some v ∈ X τ <u , or c = d ↓ for some d ∈ X Tps τ ≥u . The latter case readily leads to a contradiction, since by claim (6) in Proposition 8.2 τ (d) = u, whereas claim (4) in Proposition 2.3 shows τ (d) = d > c ≥ u. In the former case, by Table 3 (3,3) , the idempotency of d implies . Since X τ <u = X gr holds by claim (4) in Lemma 6.2, it holds that v is the inverse of v, and hence β(v ) is the inverse of β(v) in β(X). Therefore, These conclude the proof of the second statement.
3. Referring to Lemma 6.2, let δ = (β| Xgr , µ) denote the isomorphism from or in a more detailed form, Then α is an isomorphism from X to Y Z V ← ← × ker β . Indeed, intuitively, what γ•β does is that it makes each component together with its extremals in X collapse into a singleton, and also makes each coherent pair of pseudo extremals, that is, a gap {x ↓ , x} (x ↓ < x ∈ X  (2) and Definitions 6.1, 9.1, and 4.2/A. Thus, α is an order-isomorphism from × ker β . A straightforward verification using Definition 4.2/A and Table 3 shows the preservation of the monoidal operation under α. Moreover, (6.4), (7.3) and claim (4) in Proposition 8.2 shows the preservation of the residual complement, under α. In complete analogy to the way we proved the preservation of the implication under β in (6.5), we can verify the preservation of the implication under α, too. Finally, it is clear that α maps the unit element of X to the unit element of Q.E.D.
10 One-step decomposition -when u is not idempotent In order to handle the case when u is not idempotent, we shall consider a residuated subsemigroup X τ ≥u of X, which is an odd involutive FL e -algebra, albeit not a subalgebra of X, since its unit element u and residual complement u will differ from those of X. Then we prove that the group-part of X τ ≥u (namely X T τ ≥u ) is discretely embedded into X τ ≥u , that X Tc τ ≥u ≤ X T τ ≥u , and finally we will recover X, up to isomorphism, as a type IV partial lexicographic product of X τ ≥u , X Tc τ ≥u and ker β .
Proposition 10.1 Let X = (X, ≤, * •, → * • , t, f ) be an odd involutive FL e -chain with residual complement , such that there exists u, the smallest strictly positive idempotent. Assume u is not idempotent.
1. The following product table holds for * • (in the bottom-right entry the formula on the left applies if x * • y > (x * • y) ↓ ∈ X τ ≥u , whereas the formula on the right applies if x * • y > (x * • y) ↓ ∈ X τ <u or x * • y = (x * • y) ↓ .
3. X T τ ≥u is discretely embedded into X τ ≥u .

Proof
(1) Grey items in Table 4 are either readily seen by symmetry of * • or are inherited from Table 2, so it suffices to prove the items in black.
Let x ∈ X T τ ≥u . Then x ∈ X τ ≥u by Lemma 5.4. If x / ∈ X T τ ≥u then by (8.1), which, by the claim would yield x * • u = x, contradicting to x ∈ X T τ ≥u by (8.1).
(3) We prove that X T τ ≥u is closed under ⇓ and ⇑ , and neither ⇓ nor ⇑ have fixed point in X T τ ≥u . By (7.3), for w ∈ X τ <u , ) holds by (7.3), and by claim (2) it is in X T τ ≥u . Therefore, it remains to prove x ↓ ∈ X T τ ≥u if x ∈ X Tps τ ≥u . The opposite, that is, , a contradiction to Proposition 5.3.
It is straightforward that ⇓ has no fixed point in X T τ ≥u . The statements on ⇑ follow from (4.1). Q.E.D.
Lemma 10.2 Let X = (X, ≤, * •, → * • , t, f ) be an odd involutive FL e -chain with residual complement , such that there exists u, the smallest strictly positive idempotent, and u is not idempotent.
1. X τ ≥u = (X τ ≥u , ≤, * •, → * • , u, u) is an odd involutive FL e -chain with residual complement The set of positive idempotent elements of X τ ≥u equals the set of positive idempotent elements of X deprived of t.

Proof
(1) The set X τ ≥u is nonempty since by claim (4) in Proposition 2.3, u is an element in it. X τ ≥u is closed under * • and by Lemma 5.4, and hence under → * • , too, by (2.1). Clearly, u is the unit element of * • over X τ ≥u . The residual complement operation in X τ ≥u is u , since by (2.1) and (8.1), for x ∈ X τ ≥u it holds true that x → * • u = (x * • u ) = (x * • ⊥ [t] ) = x u . Next we prove that X τ ≥u is involutive, that is, that u is an order reversing involution of X τ ≥u . Indeed, if x ∈ X τ ≥u \ X T τ ≥u then x u = x ∈ X τ ≥u \ X T τ ≥u by Lemma 5.4 and claim (2), and hence (x (2) and (3) in Proposition 10.1, and using claim (3) in Proposition 10.1, it follows that (x u ) u = (x ⇑ ) ⇑ = x, so we are done. Next, u u = (u ⇓ ) = (u ) = u, so X τ ≥u is odd. Finally, for any idempotent element c such that c ≥ u, also c ∈ X τ ≥u holds because of claim (4) in Proposition 2.3. Since u is the smallest strictly positive idempotent element in X, the same claim shows that the only positive idempotent element of X which is missing from X τ <u is t.
(2) Claim: u is inverse operation on X T τ ≥u . Indeed, first let x ∈ X Tc τ ≥u . By (7.3) and Table 1 (3,3) On the other hand, since x > x ↓ , by residuation and by using that X is a chain, x * • x u = x * • (x ↓ ) > t holds, and it implies x * • x u ≥ u, since x * • x u ∈ X τ ≥u by Lemma 5.4, and u is the smallest element of X τ ≥u , which is greater than t. This settles the claim.
Therefore, X T τ ≥u ⊆ (X τ ≥u ) gr holds. If, by contradiction, there would exists p ∈ (X τ ≥u ) gr \ X T τ ≥u then its inverse is p u by Theorem 2.4, and by the claim, p u would also be in (X τ ≥u ) gr \ X T τ ≥u . Thus p * • p u = u ∈ X T τ ≥u follows, a contradiction to Table 2 (4,4) .
As for the second statement, Table 4 (1,1) shows that X Tc τ ≥u is closed under * •.
α is clearly injective and surjective.
Summing up, we have verified α(x ) = α(x) for all x ∈ X.

Group representation
The main theorem of the paper asserts that up to isomorphism, any odd involutive FL e -chain which has only finitely many positive idempotent elements can be built by iterating finitely many times the type III and type IV partial lexicographic product constructions using only linearly ordered abelian groups, as building blocks.
Remark 11.2 Note that by Lemmas 9.4 and 10.2, and claim (4) in Theorem 2.4, linearly ordered abelian groups are exactly the indecomposable algebras with respect to the type III and type IV partial lexicographic product constructions.
Remark 11.3 Denote the one-element odd involutive FL e -algebra by 1. If the algebra in Theorem 11.1 is bounded then in its group representation G 1 = 1, since all other linearly ordered groups are infinite and unbounded, and both type III and type IV constructions preserve boundedness of the first component.
We say that an odd involutive FL e -chain X is represented as a finite (or more precisely, n-ary) partial lexicographic product of linearly ordered abelian groups G 1 . . . , G n , if X arises via finitely many iterations of the type III and type IV constructions using linearly ordered abelian groups G 1 . . . , G n in the way it is described in Theorem 11.1. By using this terminology we may rephrase Theorem 11.1: Every odd involutive FL e -chain, which has only finitely many positive idempotent elements can be represented as a finite partial lexicographic product of linearly ordered abelian groups. By Hahn's theorem one can embed the linearly ordered abelian groups into some lexicographic products of real groups. Therefore, a side result of Theorem 11.1 is the generalization of Hahn's embedding theorem from the class of linearly ordered abelian groups to the class of odd involutive FL e -chains which have only finitely many positive idempotent elements: Corollary 11.4 Odd involutive FL e -chains which have exactly n ∈ N positive idempotent elements embed in some n-ary partial lexicographic product of lexicographic products of real groups.
Finally, we remark that ordinary lexicographic products can be used instead of partial lexicographic products if the less ambitious goal of embedding only the monoidal reduct is aimed at.
Corollary 11.5 The monoid reduct of any odd involutive FL e -chain which has only finitely many idempotent elements embeds in the finite lexicographic product where G 1 , G 2 , . . . , G n are the linearly ordered abelian groups of its group representation.
Proof By Theorem 4.4, Observe that the monoidal reduct of X ← × G embeds into the monoidal reduct of X (Xgr) ← ← × G, which in turn embeds into the monoidal reduct of X ⊥ ← ×G ⊥ . Now let an odd involutive FL e -chain, which has only finitely many idempotent elements, be given. Take its group representation. Guided by its consecutive iterative steps, in each step consider ( In the end, this results in the original algebra being embedded into the lexicographic product G ⊥ where G 1 , G 2 , . . . , G n are linearly ordered abelian groups. Q.E.D.