Group representation for even and odd involutive commutative residuated chains

For odd and for even involutive, commutative residuated chains a representation theorem is presented in this paper by means of direct systems of abelian o-groups equipped with further structure. This generalizes the corresponding result of J. M. Dunn about finite Sugihara monoids.


Introduction
J. M. Dunn has described the structure of finite Sugihara monoids by showing in [2] that subdirectly irreducible Sugihara monoids are linearly ordered and by describing the structure of finite Sugihara chains in [15]. We generalize Dunn's results by dropping the idempotence and the finiteness axioms.
In line with the tradition of representing classes of residuated lattices by simpler and better known structures such as groups or Boolean algebras, Mundici's celebrated categorical equivalence theorem represents MV-algebras -a variety which corresponds to Lukasiewicz logic L [12] -by ℓ-groups with strong units using a truncation construction [43]. By dropping the divisibility axiom of MV-algebras one obtains the variety of IMTL-algebras which correspond to the logic IMTL [14,45,17,9,37,16,44,10,5,13,24]. The non-integral analogue of the class of IMTL-algebras shall be represented by o-groups in this paper. By replacing the integrality axiom of IMTL-algebras by one of its two natural non-integral analogues one obtains the class of odd and the class of even involutive semilinear FL e -algebras, the former of which is also known as the class of IUL f p -algebras corresponding to the Involutive Uninorm Logics with fixed point (IUL f p ) introduced by G. Metcalfe in [38], see also [39]. We shall represent all subdirectly irreducible members of these two classes by means of abelian o-groups. Subdirectly irreducible even or odd involutive FL e -algebras are linearly ordered. Little is known about the structure of IMTL-algebras and an effective structural description for this class seems to be out of reach at present, arguably, such a structural description is impossible. However, as shown in this paper, the non-integral counterpart of the class of IMTL-algebras possesses a representation by groups, similar to MV-algebras or to cancellative commutative residuated lattices [41].
Whether Involutive Uninorm Logic (IUL) is standard complete or not is a problem posed by G. Metcalfe and F. Montagna [39] and has become a long-standing open problem which has invoked considerable effort to solve it with no avail 1 . The representation theorem in this paper may serve as a step toward settling the problem of standard completeness of IUL in an algebraic manner.
Group theory is the science of symmetries whereas the theory of inverse semigroups is the science of partial symmetries [35]. Our main result is reminiscent to the representation theorem of a subclass of inverse semigroups, called Clifford semigroups. Every Clifford semigroup is isomorphic to a strong semilattice of groups [35,Theorem 12 in Section 5.2] (see also [51]), and the starting point of the strong semilattice construction is a direct system of groups indexed by a semilattice. In our case the starting point is a direct system of abelian o-groups indexed by a chain and equipped with some extra structure. The definition of the product in the strong semilattice construction is of the same P lonka sum fashion (see [48]) as in our construction in (4.8). However, in our case we need to modify the abelian ogroups prior to applying (4.8), and we also need to handle the ordering, the residual operation, the isotonicity of the product with respect to the ordering etc. In particular, we introduce in this paper an ordering on P lonka sums, called the directed lexicographic order. P lonka sums have found applications not only in computer science, in particular in the theory of program semantics [49], but recently its corresponding regular varieties have been unexpectedly connected to logic. They provide algebraic semantics to logics obtained by imposing a deductive filter to other logics; for instance, Paraconsistent Weak Kleene logic coincides with the regularization of the variety of Boolean algebras [7,8].
Integral residuated lattices has been widely studied in the literature [1,4,6,12,22,25,34]. However, as noted by N. Galatos and J. G. Raftery in [20], non-integral residuated structures and consequently, substructural logics without the weakening rule are less understood than their integral counterparts. To overcome this, they established category equivalences to carry over the algebraic knowledge on integral structures to non-integral ones [20,21]. These kind of categorical approaches use some algebraic background knowledge which is available if idempotency is postulated. Our paper makes a step in the direction of providing an algebraic insight into the non-idempotent, non-integral, non-divisible setting.
Prominent examples of odd involutive FL e -algebras are abelian ℓ-groups [3] and odd Sugihara monoids. These two classes of algebras represent two extremities: there is a single idempotent element in any lattice-ordered abelian group, whereas all elements are idempotent in any odd Sugihara monoid. The former class constitutes an algebraic semantics of Abelian Logic [11,40,47] while the latter constitutes an algebraic semantics of IUML * , which is a logic at the intersection of relevance logic and many-valued logic [20]. Our representation theorem puts these two rather distant logics (and more) under the same methodological umbrella.
Despite the extensive literature devoted to classes of residuated lattices, there are still few effective structural descriptions. All these results in [1,12,22,25,33,34,36,42,23,50,46] (and also others where the focus is not a structural description in the first place such as in [20,21] for example) postulate semilinearity which renders the subdirectly irreducible members linearly ordered, and either integrality together with the naturally ordered condition 2 or idempotency. Our study contributes to the structural description of residuated lattices which are semilinear but neither integral nor naturally ordered nor idempotent; a first such effective structural description to the best of our knowledge besides [32,30].
A representation theorem has been presented in [32,30] for those odd involutive FL e -chains where the number of idempotent elements of the algebra is finite by means of partial sublex products of abelian o-groups, which are well understood mathematical objects that are much more regular than what had been expected to need for describing these particular FL e -chains. In the present paper we prove a representation theorem for both even and odd involutive FL e -chains without assuming any constrains on the set of their idempotent elements. The conic case is also settled. While the construction in the representation of [32,30] is done by starting with an abelian o-group and iteratively enlarging it by other abelian o-groups until the obtained structure becomes isomorphic to the given algebra, here we present a structural description using direct systems, without referring to iteration. To this end, a core auxiliary result concerns a one-to-one correspondence between two lattice ordered classes: odd involutive FL e -algebras and even involutive FL e -algebras with an idempotent falsum constant. Here we make use of and further develop ideas of E. Casari [11], see Section 6.
Since ℓ-groups are very specific mathematical objects compared to residuated lattices, having a representation of a class of residuated lattices by subclasses of ℓ-groups is a rarity. It is reasonable to expect that the inherent larger complexity of the more general classes of residuated lattices renders such a representation, if exists, quite involved in its constructional part. This phenomenon can be observed in [30,32], for example, where the notions of two different partial sublex product constructions together with the technique of iteration have been used to describe the class of residuated lattices which is in our focus in the present paper with the additional assumption that the number of idempotent elements of the algebra is finite. In the present paper the complexity of the residuated structure is coded in the system of homomorphisms and in the delicate way of constructing the algebra from the direct system. Applications of the main result of this paper to amalgamation and densification problems are foreshadowed in [27].

Preliminaries
An FL e -algebra is a structure (X, ∧, ∨, ·, →, t, f ) such that (X, ∧, ∨) is a lattice, (X, ≤, ·, t) is a commutative, residuated monoid (the unit element t is also referred to as the truth constant), and f is an arbitrary constant, called the falsum constant. Being residuated means that there exists a binary operation →, called the residual operation of ·, such that xy ≤ z if and only if x → z ≥ y. This equivalence is called adjointness condition, (·, →) is called an adjoint pair. Equivalently, for any x, z, the set {v | xv ≤ z} has its greatest element, and x → z, the residuum of x and z, is defined as this element: x → z := max{v | xv ≤ z}; this is called the residuation condition. Being residuated implies that · is increasing. One defines the residual complement operation by x ′ = x → f and calls an FL e -algebra involutive if (x ′ ) ′ = x holds. An involutive FL e -algebra is called odd if the residual complement operation leaves the unit element fixed, that is, t ′ = t, and even if the following (two) quasi-identities hold: x < t ⇔ x ≤ f . The former condition is equivalent to f = t, while the latter quasi-identities are equivalent to assuming that f is the lower cover of t (and t is the upper cover of f ) if chains (or more generally conic algebras) are considered. An FL e -algebra is called integral if the unit element of the multiplication is the top element of its universe. Commutative residuated lattices are the f -free reducts of FL e -algebras. The geometric meaning of t ′ = f , which is valid in all involutive FL e -algebras, is that the two constants are positioned symmetrically inside the underlying set. Therefore, one extremal setting is the integral case, when t and f are the top and bottom elements of X, respectively, and the other extremal one is the even or odd case when the two constants are both "in the middle of X". A bottom-top setup is not possible: if a residuated lattice has a bottom element then that element is annihilating, hence cannot be the unit of the multiplication 3 . Both odd or even involutive FL e -chains and IMTL-chains are involutive residuated chains with a single additional postulate: the integrality condition of IMTL-chains postulates the unit element to be in one of its possible extremal positions (top), whereas the odd or even condition postulates the unit element to be in its other extremal position (in the middle).
Section 3 contains results on the local unit element function of involutive FL ealgebras, a key concept for our representation theorem. Even involutive FL ealgebras with non-idempotent and idempotent falsum constants will be characterized with respect to odd involutive FL e -algebras in Sections 5 and 6, respectively. In Section 4 even and odd involutive FL e -chains will be partitioned into their socalled layer algebras, which are also even or odd involutive FL e -chains but more specific ones than the original algebra in the sense that they are either cancellative or are close to being such: in the latter case there exists a canonical homomorphism which maps the layer algebra into a cancellative one from which the layer algebra can be uniquely recovered. This specificity allows for establishing a connection between them and abelian o-groups in Section 7, by using the characterizations of Sections 5 and 6. These lead to the main result of the paper in Section 8: a one-to-one correspondence in a constructive manner between the class of all even or odd involutive commutative residuated chains and the class of bunches of layer groups. Layer groups are direct systems of abelian o-groups equipped with further structure.

The local unit element function
Let (X, ≤) be a poset. For x, y ∈ X we say that y is a cover of x if y > x and there exists no z ∈ X such that y > z > x. For x ∈ X let x ↑ be the unique cover of x if such exists, and let x ↑ = x otherwise. Define x ↓ dually. Call ↓ and ↑ the neighbour operations of (X, ≤). A partially ordered algebra with a poset reduct is called discretely ordered if for any element x, x ↓ < x < x ↑ holds. If ′ is an order-reversing involution of X then it holds true that (3.1) x ′ ↑ = (x ↓ ) ′ and x ′ ↓ = (x ↑ ) ′ . Algebras will be denoted by bold capital letters, their underlying sets by the same regular letter unless otherwise stated. Sometimes the lattice operators of an FL e -algebra will be replaced by their induced ordering ≤ in the signature, in particular, if an FL e -chain is considered, that is, if the ordering is total. Call the elements x ≥ t of an FL e -algebra X = (X, ∧, ∨, ·, →, t, f ) positive. Call X conic if all elements of X are comparable with t. Assume X is involutive. For x ∈ X let or equivalently, define τ (x) to be the greatest element of Stab x = {z ∈ X | zx = x}.
A key step toward our representation theorem is to understand the role of the τ function. In investigating more specific odd involutive FL e -algebras τ (x) was recognized to play the role of the 'absolute value of x' and was denoted by |x|, see [50]. The definition of layers in (3.2) and Lemma 3.1 reveal the true nature of τ in the present, more general setting: τ can be view as a 'local unit element' function. For any positive idempotent element u define and call it the u-layer of X. The X u 's form a partition of X by claim (vii) below, and if x ∈ X u then τ (x) is the unit element for the subset X u of X by claim (v) 4 .
Corresponding to the rotation-annihilation construction the idea of decomposing a class of FL e -algebras with the help of a closely related notion, the skeleton function of the algebra 5 has been presented already in [28]. The name local unit function and the corresponding idea of using it to localize elements of the algebra has been introduced in [32], and the way of localizing as done in (3.2) is being introduced in the present paper, see also [31]. Some of the statements of the following lemma can be found in [32] or in [19], too. We include their proofs to keep the paper self-contained. As usual, when multiplication is denoted by ·, we write xy instead of x·y.
if the algebra is conic and either even or t ≤ f then y 1 > y implies ( x ∈ X} is equal to the set of positive idempotent elements of X, (viii) x 1 y > xy holds whenever x 1 > x and y is invertible, if X is odd then the (X, ∧, ∨, ·, t)-reduct of X is a lattice-ordered abelian group if and only if · is cancellative 6 , (xii) for x ≥ t, τ (x) ≤ x holds, 4 For a first glance it might occur to the reader that the definition of Xu is equivalent to the well-known way of localizing in semigroup theory, namely, for an idempotent u, to the universe Mu = {x ∈ X : xu = x} of the greatest unitary subsemigroup of X with unit u. It is not the case if u > t. Rather, Xu can be regarded as the "outer edge" of Mu in the following sense: it holds true that Mu = v≤u Xv and Xu = Mu \ v<u Xv (u and v are positive idempotents). 5 The skeleton function was defined there as the residual complement of what we coined localunit function. 6 When we (loosely) speak about a subgroup of odd involutive FLe-algebra X in the sequel, we shall mean a cancellative subalgebra of X.
(xiii) if X is an odd chain then X t contains all the invertible elements of X, and it is a universe of an abelian o-group, its inverse operation is the residual complement operation. (xiv) if X is an odd or even, cancellative, discretely ordered chain then zt ↓ = z ↓ , (xv) if X is an odd or even chain then xy < x 1 y 1 holds whenever x < x 1 and y < y 1 , (xvi) if X is an odd or even chain and A is an X-term which contains only the operations ·, → and ′ then for any evaluation e of the variables of A into X, τ (e(A)) equals the maximum of the τ -values of the variables and constants of A under e,

Proof.
(i) Obvious by residuation. (ii) Using that ′ is an involution one obtains (xy Since the algebra is conic and involutive, every element is comparable with f , too. Indeed, if for any a ∈ X, a were not comparable with f then, since ′ is an order reversing involution, a ′ were not comparable with f ′ , and f ′ = t since the algebra is involutive, a contradiction. Therefore, by residuation, Likewise, if the algebra is even then y 1 y ′ ≤ f implies y 1 y ′ < t, that is, y 1 y ′ ≥ t, since the algebra is conic. Therefore, Since t is the unit element, Stab x is nonempty. Therefore, by residuation τ (x) = x → x is its largest element, hence τ (x)x = x holds. Since tx = x, τ (x) ≥ t follows by residuation. (vi) If u ≥ t is idempotent then from uu = u, u → u ≥ u follows by adjointness.
But for any z > u, uz ≥ tz = z > u, hence τ (u) = u follows. On the other hand, by claim (v), τ (u) = u implies u ≥ t, and also the idempotency of u follows since uu = uτ (u) = u. (vii) If u > t is idempotent then claim (vi) shows that u is in the range of τ . If u is in the range of τ , that is τ (x) = u for some x ∈ X then if τ (τ (x)) = τ (x) then it implies τ (u) = u, hence u is a positive idempotent element by claim (vi), and we are done. Hence it suffices to prove τ (τ (x)) = τ (x) for all x. By claim (ii), (viii) x 1 y ≥ xy holds by monotonicity, and if x 1 y = xy then x 1 = x 1 yy −1 = xyy −1 = x, a contradiction. (ix) By claim (ii) and the involutivity of Since · is cancellative, the strictly increasing nature of · clearly follows: if u < v then uw < vw. Therefore, xa > xt = x for any a > t, and hence x → x = t. An application of claim (ii) ends the proof. (xi) Necessity is straightforward, sufficiency follows from claim (x) since f = t.
(xiii) If x ∈ X t then τ (x) = x → x = t and hence xx ′ (ii) = t ′ (i) = f = t holds since the algebra is odd. This shows that all elements of X t are invertible in X and the inverse operation is ′ . On the other hand, if x ∈ X is invertible, that is, xy = t for some y ∈ X then τ (x) = t: for every z > t it holds true that xz > x since the opposite, that is, xz = x would imply z = tz = (yx)z = y(xz) = yx = t, a contradiction. Invertible element are clearly closed under · and ′ . Since the order is total, the meet and the join of two invertible elements are also invertible. (xiv) Since · is cancellative, · is strictly increasing; we shall use it without further mention. Now, zt ↓ < zt = z holds since t ↓ < t follows from the algebra being discretely ordered. Contrary to the statement, assume that there exists a such that zt ↓ < a < z. Multiplying with z ′ , f t ↓ < az ′ < f follows by claim (x). In the odd case it yields t ↓ < az ′ < t, a contradiction to the definition of ↓ . In the even case it yields Assume the opposite, which is In the odd case these reduce to ( In the even case these reduce to (x ′ x 1 )(y ′ y 1 ) < t and x ′ x 1 ≥ t. Analogously we obtain y ′ y 1 ≥ t. Therefore (x ′ x 1 )(y ′ y 1 ) ≥ tt = t follows, a contradiction. (xvi) We have already seen τ (x) = τ (x ′ ) in claim (ix).
Next, we claim τ (xy) = max(τ (x), τ (y))(= τ (x)τ (y) 7 ). Indeed, τ (xy) ≥ τ (x) holds by residuation since xy→xy ≥ x→x is equivalent to yx(x → x) ≤ xy. Assume z := τ (xy) > max(τ (x), τ (y)). Since τ assigns to x the greatest element of the stabilizer set of x, therefore z > max(τ (x), τ (y)) implies that z does not stabilize x neither y, hence x < zx and y < zy holds by the monotonicity of ·. On the other hand, (zx)(zy) = ((xy)z)z (v) = xy follows, a contradiction to claim (xv). This settles the claim. By claim (ii), any term which contains only the connectives ·, → and ′ can be represented by an equivalent term using the same variables and constants but containing only · and ′ . An easy induction on the recursive structure of this equivalent term using the two claims above concludes the proof.
The next lemma states that for an odd involutive FL e -algebra, it is exactly cancellativity which is needed to make it a lattice ordered group. (1) For a cancellative odd involutive FL e -algebra with residual complement ′ , λ(X) = (X, ∧, ∨, ·, −1 , t) is a lattice-ordered abelian group, called the lattice-ordered abelian group induced by X, where (2) For a lattice-ordered abelian group With the above assumptions it holds true that ι(λ(X)) = X and λ(ι(G)) = G.
Proof. Claim (xi) in Lemma 3.1 confirms the first statement, the rest is folklore or obvious.
The following lemma will simplify the proof of Theorem 6.3 and Lemma 4.2.
Lemma 3.3. If there exists a dualizing element c of M then · is residuated and its residual operation is given by x → y = (x(y → c)) → c.
Proof. zx ≤ y is equivalent to zx ≤ (y → c) → c. By adjointness it is equivalent to (zx)(y → c) ≤ c. By associativity it is equivalent to z(x(y → c)) ≤ c, which is equivalent to z ≤ (x(y → c)) → c by adjointness. By residuation x → y = (x(y → c)) → c follows.

Odd and even involutive FL e -chains vs. bunches of layer algebras
We shall prove the main theorem of the paper for three different kinds of involutive FL e -chains: for odd involutive FL e -chains, for even involutive FL e -chains with an idempotent falsum, and for even involutive FL e -chains with a non-idempotent falsum.
Definition 4.1. Let (κ, ≤ κ ) be a totally ordered set with least element t, and let an ordered triple κ I ,κ J , {t} be a partition of κ, whereκ I andκ J can also be empty. Define κ o , κ J , and κ I by one of the rows of Table 1 10 .
) be a family of involutive FL e -chains indexed by elements of κ (let ′ u denote the residual complement operation, ↓u and ↑u the 8 Dualizing elements have been defined only in residuated structures in the literature, see e.g. [19,Section 3.4.17.]. 9 That is, the exists the greatest element of the set {z ∈ M | xz ≤ c}. 10 Explanation: if the first, the second, or the third row is used then the algebra corresponding to the bunch will be odd, even with a non-idempotent falsum constant, or even with an idempotent falsum constant, respectively. Table 1.
and such that for Call A = X u , ρ u→v κo,κJ ,κI ,≤κ a bunch of layer algebras. Call the X u 's the layer algebras, call κ, ≤ κ the skeleton, call κ o , κ J , κ I the partition of the skeleton, and call X u , ρ u→v κ the direct system of layer algebras over κ. Note that κ can be recovered from its partition, (and ultimately, from We prove that every odd or even involutive FL e -chain can be represented by a unique bunch of layer algebras. Later, in Section 7 we prove that every bunch of layer algebras can be represented by a unique bunch of layer groups. (1) Given an odd or an even involutive FL e -chain X = (X, ≤, ·, →, t, f ) with residual complement operation ′ , is a bunch of layer algebras, called the bunch of layer algebras of X, where Table 2, Table 2.
if X is even and f is idempotent 11 Hence with a non-idempotent falsum. 12 If u ∈ κo then ρu→v(u) = ρu→v(u ′ u ) trivially holds since (2) Given a bunch of layer algebras is an involutive FL e -chain, called the involutive FL e -chain derived from A, where for short, for x ∈ X u and y ∈ X v , 13 and t is the least element of κ. X A is odd if t ∈ κ o , even with a nonidempotent falsum if t ∈ κ J , and even with an idempotent falsum if t ∈ κ I . (3) For a bunch of layer algebras A, A (XA) = A, and for an odd or even involutive FL e -chain X, X AX = X.
Proof. (1): κ is the set of positive idempotent elements of X by claim (vii) in Lemma 3.1. Therefore, the least element of κ is t, and κ, being a subset of X, is totally ordered. The ordered triple κ I ,κ J , {t} is clearly a partition of κ, wherē κ I andκ J can also be empty. Let u ∈ κ. X u is nonempty since u ∈ X u holds by claim (vi) in Lemma 3.1, and, being a subset of X, X u is totally ordered by ≤ u . X u is closed under · u , → u , and ′ by claim (xvi) in Lemma 3.1, and thus Note that for u, v ∈ κ, uv = maxκ(u, v) since u, v are positive idempotents. 14 Alternatively, we may write x ≤ y iff ρuv(x) ≤uv ρuv(y) except if u >κ v and ρuv(x) = ρuv(y) Next we prove that the X u 's satisfy (4.1).
: If u ∈ κ o then X is odd by Table 2. By claim (xiii) in Lemma 3.1 and by Lemma 3.2, X u is a cancellative odd involutive FL e -chain. : If u ∈ κ J then by Table 2, u = t or u ∈κ J . In both cases u ′ is not idempotent.
We prove that X u is discretely ordered by showing where ↓u denotes the neighbour operation on X u . It holds true that Table 2 shows that X is even. Therefore, by denoting ≤ u ′ t = u ′ and equality cannot not hold since u ′ is not idempotent. Now xu ′ (4.14) ≤ xt = x follows. Assume, by contradiction xu ′ = x. It would imply xy = x(u ′ u ′ ) = (xu ′ )u ′ = x, hence by claim (iii) in Lemma 3.1, (x ′ y ′ ) ′ ≥ xy = x, and in turn x ′ y ′ ≤ x ′ would follow on the one hand. On the other hand, from y ′ (4. 15) > u ≥ t, by monotonicity x ′ y ′ ≥ (4.15). We have just seen that xu ′ < x. Next, assume that there exists z ∈ X u such that xu ′ < z < x holds. Since z < x, xu ′ ≥ (z ′ u) ′ = z ′′ = z follows by claim (iv) in Lemma 3.1, a contradiction, so (4.13) is confirmed.
Next we show that X u is even: Summing up, X u is a discretely ordered cancellative even involutive FL echain.
: If u ∈ κ I then by Table 2, u = t or u ∈κ I . In both cases u ′ is idempotent. We show that X u is even: first, u ′ u < u holds since if u ∈κ I then u > t and then X is even by the third row of Table 2 and u ′ u (4.11) = u ′ = t ′ L3.1(i) = f X is even < t = u, and second, by claims (xii) and (xvi) in Lemma 3.1, no element x ∈ X such that u ′ < x < u can be in X u (if u ′ < x < u then u ′ < x ′ < u, so we may safely assume x ≥ t, and then x ∈ X u implies u = τ (x) ≤ x < u, a contradiction); thus u ′ u = u ↓u . Summing up, X u is an even involutive FL e -chain with an idempotent falsum. It remains to Next we prove that ρ u→v is a homomorphism from the residuated lattice reduct of X u to the residuated lattice reduct of X v . Let u, v ∈ X be positive idempotent elements of X such that u < v. ρ u→v maps X u to X v by claim (xvi) in Lemma 3.1. ρ u→v preserves the ordering since · is monotone. ρ u→v preserves products since · is associative and v is idempotent. To show that ρ u→v preserves the residual operation we proceed as follows. Let and hence v does not stabilize x, x < vx follows. Therefore, x ′ (vy) Therefore, ρ u→v (x → u y) Finally, ρ u→v preserves the unit element: Summing up, ρ u→v is a homomorphism from the residuated lattice reduct of X u to the residuated lattice reduct of X v .
To conclude the proof of claim (1) it only remains to prove (A1): For u, v, w ∈ X positive idempotent elements such that u ≤ v ≤ w, and  = ρ u→v (u).
(2): We conclude by a series of claims. ≤ is a total ordering on X.
• Irreflexivity of < is immediate form the irreflexivity of < κ .
• For x ∈ X u , tx = u · u x = x holds using that u is the unit element of X u .
′ is an order reversing bijection on X.
We start with two claims.
and is equal to and we are done. Now, multiplying both sides by If v ∈ κ J then by (4.1) we can apply claim (xiv) in , so we are done. If v ∈ κ I then by (4.1), X v is an even involutive FL e -chain with an idempotent falsum. Therefore, X v = Sp(A, H) by Theorem 6.3, and is an element of H by claim (1c) in Theorem 6.3. On the other hand, v ↓v is clearly in H • . Therefore, • Since for u ∈ κ, ′ u is of order 2, so is ′ over X by (4.9), hence ′ is a bijection. It remains to prove that ′ is order reversing. Let X u ∋ x ≤ y ∈ X v . If u = v then x ≤ u y holds by (4.7), hence y ′ u ≤ u x ′ u follows since X u is involutive, thus y ′ ≤ x ′ holds by (4.7) and (A1).
• It suffices to prove that for x ∈ X, there exists x → t ′ and (4.17) Since · is commutative and since ′ is an order reversing bijection, we may , and by (4.7) it is equivalent to x ≤ y ′ v (4.9) = y ′ . Summing up, we have shown that (X, ≤) is a chain and (X, ·, t) is a commutative monoid. Since t ′ is a dualizing element of (X, ≤, ), Lemma 3.3 shows that (X, ≤, ) is residuated and x → y = (xy ′ ) ′ . Since ′ (given in (4.9)) coincides with the residual complement of X (given by x → t ′ , see (4.17)), and since ′ is an order reserving involution on X, it follows that X is involutive. Finally, by (4.1), X t and hence also X is odd if t ∈ κ o , X is even with a non-idempotent falsum if t ∈ κ J , and X is even with an idempotent falsum if t ∈ κ I .
(3): Let A = X u , ρ u→v κ be a bunch of layer algebras, and adapt the notations in Definition 4.1 and the definitions in claim (2). To see that the universe of the u th layer algebra of A (XA) is equal to X u which is the universe of the u th layer algebra X u of A, we need to prove that for x ∈ X (where X is given in (4.5)), x → x = u if and only if x is in X u , the universe of the u th -layer algebra: x ∈ X implies that x ∈ X v for some v ∈ κ. Now u = x→ x = where the last equality holds since X v is closed under ′ v and · v and thus x · v x ′ v is in X v . Therefore, u ∈ X v follows. Hence u = v must hold since u ∈ X u and X is the disjoint union of the X u 's by (4.5). The definition of the ordering relation in claim (1) and (4.7) show that the ordering of the u th layer algebra of A (XA) is the same as the ordering ≤ u of the u th layer algebra X u of A. Likewise show (4.8) and the definition of the monoidal operation in claim (1) that the monoidal operation of the u th layer algebra of A (XA) is the same as the monoidal operation · u of the u th layer algebra X u of A. Since both the u th layer algebra of A (XA) and the u th layer algebra X u of A are involutive FL e -chains over the same universe, equipped with the same ordering relation and the same product operation, their residual operations -which are uniquely determined by these -must coincide, too. The unit element of X u is u, therefore u acts as the unit element of the u th layer algebra of A (XA) (which is over the same set X u and is equipped with the same monoidal operation, as we have seen above). Finally, the falsum constant of X u is u ′ u . On the other hand, the falsum constant of the u th layer algebra of A (XA) is where the last equality holds by residuation since u is the unit element over X u . Summing up, A (XA) = A.
Let X = (X, ≤, ·, →, t, f ) be an odd or even involutive FL e -chain, and adapt the definitions in claim (1). We have seen in the proof of claim (1) that the X u 's are nonempty. It is straightforward that they are disjoint, too, and their union, which is the universe of X AX , see (4.5), is equal to X. To prove that the ordering of X and of X AX coincide, first we prove the following statement. If v < u and x ∈ X v then (4. 18) ρ u (x) = min{z ∈ X u : z ≥ x} > x.
Indeed, ux ∈ X u by claim (xvi) in Lemma 3.1, and ux ≥ x holds since ux ≥ tx = x. By contradiction, assume that there exists z ∈ X u such that x < z < ux. Since ≤ is total, by adjointness z ′′ < ux is equivalent to f < z ′ (ux) = (z ′ u)x z ′ ∈Xu = z ′ x. Finally, since ≤ is total, by adjointness f < z ′ x is equivalent to x > z ′′ = z, a contradiction. Referring to (4.18) a moment's reflection shows that the ordering ≤ of X coincides with the ordering of X AX given by (4.7). Since for x ∈ X u and y ∈ (uux)(vvy) = ((uv)x)((uv)y) = ρ uv (x)ρ uv (y) = ρ uv (x) · uv ρ uv (y), the monoidal operation · (of X) coincides with · given in (4.8). Since both X and X AX are involutive FL e -chains over the same universe, equipped with the same ordering relation and the same product operation, their residual operations -which are uniquely determined by these -must coincide, too. By (4.3), the unit element of X t is t, and hence the unit element of X AX is also t. Finally, the falsum constant = f . Therefore, the falsum constant of X AX is also t ′ = f . Summing up, X AX = X. Remark 4.3. Two elements of the construction of X A in Lemma 4.2/(2) are reminiscent to the P lonka sum construction [48]. In the P lonka sum construction the universe of the algebra is the disjoint union of the universes of the algebras in the direct system. So is in our construction, c.f. (4.5). In the P lonka sum construction every binary operator • ∈ {·, →, ∧, ∨} would be given in terms of the direct system as follows: for x ∈ X u , y ∈ X v , (4.19) x • y = ρ u→uv (x) • uv ρ u→uv (y) In our construction it is the case only for • ∈ {·}, c.f. (4.8). As for • ∈ {→}, if → were given by (4.19) then for any positive idempotent elements u < v, it would hold But in an involutive FL e -chain it would be equivalent to vu ′ = uv ′ by Lemma 3.1/(ii), a contradiction, . Finally, notice that P lonka's definition doesn't work for • ∈ {∧, ∨} since (4.19) would render different elements of the universe equal. Indeed, if u < v are positive idempotent elements, x ∈ X u and ρ u→v (x) = y ∈ X v then according to (4.19), x ∧ y = ρ u→v (x) ∧ v y = y = ρ u→v (x) ∨ v y = x ∨ y yielding x = y, a contradiction since x and y are in different layers. Even though our definition is applied in our specific (residuated lattice) setting only, we have introduced in (4.7) a general way of ordering P lonka sums, call it the directed lexicographic order. Making use of the (slight) similarity between our construction and that of P lonka's would have made the related proof only a few lines shorter. Therefore, we have included those few lines, too, to make the treatment self-contained.
Bunches of layer algebras will, in turn, be represented by bunches of layer groups in Section 7.1. To that end, in the following two sections the necessary auxiliary results (along with some which are more general than what we actually need in this paper) will be developed.

Even involutive FL e -chains with non-idempotent falsum constants -Changing the falsum constant
An abelian o-group is called discrete, if there exists the smallest positive element greater than the unit element. It is equivalent to saying that the abelian o-group is discretely ordered, or that its induced cancellative odd involutive FL e -algebra is discretely ordered. In the following definition the residuated chain reduct of the FL e -chain is left unchanged, and only the falsum constant, and thus also the residual complement are changed slightly.
Proof. The rest being obvious we prove the involutivity of X ↓ and Y ↑ . (1): To see that X ↓ is involutive, denote the residual complement operation of X by ′ , and the inverse operation of λ(X) by −1 . Then holds, and hence (3.1) confirms involutivity.
6. Even involutive FL e -algebras with idempotent falsum constantssubgroup splits of odd involutive FL e -algebras As an investigation into the structure of residuated lattices which are not necessarily totally ordered, a one-to-one correspondence between pairs of an odd involutive FL e -algebra and a cancellative subalgebra of it, and even involutive FL ealgebras where the residual complement of the unit element is idempotent will be proved in this section.
In groups the unit element has two different roles to play. It serves as the unit element of the multiplication, and also the product of any element by its inverse is equal to it. We shall replace the unit element of any odd involutive FL e -algebra by two elements, each will inherit a single role. This way both the unit element itself and its two roles will be "split" into two. Some anticipatory examples: the unit-split algebra of the one-element group will be the two-element Boolean algebra, and the unit-split algebra of the Sugihara lattice S Z will be the Sugihara lattice S Z ⋆ , (both named by Meyer, see [2][p. 414]). Moreover, not only the unit element, but in fact any subgroup of an odd involutive FL e -algebra can be "split", that is, each element of the subgroup of an odd involutive FL e -algebra will be replaced by two elements. We prove that by splitting (and thus "doubling") a subgroup of any odd involutive FL e -algebra we obtain an even involutive FL e -algebra with an idempotent falsum constant, and each even involutive FL e -algebra with an idempotent falsum constant arises this way in a unique manner (Theorem 6.3).
Such a splitting method has roots in the literature. In [11] an analogous idea has been applied to abelian pregroups. Our method here is both more general and more specific: the algebras that we split are more general than that of [11], however, in our approach the (second) algebra we use in the splitting procedure is the two element Boolean algebra (each element is split into two elements only) whereas in [11] it is taken from the more general class of zero-pregroups. The notion of the melting structure (of a pregroup) in [11] is analogous to the image (of the even involutive FL e -algebra with idempotent falsum constant) under what we call its canonical homomorphism. The splitting procedure in [11] is precursor also for the partial lex-product construction of [32], which when further generalized to partial sublex-products has been capable to describe the structure of all odd, involutive FL e -chains which have only finitely many idempotent elements [32,30]. be given by and for x, y ∈ X, Definition 6.2. Let X = (X, ∧, ∨, ·, →, t, t) be an odd involutive FL e -algebra with the lattice ordering ≤ of X is extended to Y by letting where ↑ and ↓ denote the neighbour operations of Y, Theorem 6.3. The following statements hold true.
(1) Let X be an odd involutive FL e -algebra, H be a cancellative subalgebra of it, and Y be Sp (X, H), the H-split of X. Then (a) Y is an even FL e -algebra with an idempotent falsum constant.
(2) Let Y be an even involutive FL e -algebra with an idempotent falsum constant. Then (a) there exists a unique pair X, H of an odd involutive FL e -algebra X = (X,∧,∨, ·, →, t, t) and a cancellative subalgebra H of X such that Y is Sp (X, H), the H-split of X. X and H are given by π 1 (Y) and π 2 (Y), respectively. (Sp(X, H)).
Proof. (1a): It is obvious from (6.6) that (6.10) for a ∈ H, a is the unique cover of a • .
Therefore, by letting k : X → Y , a moment's reflection shows that ≤ Y is a lattice ordering on Y and the corresponding lattice operations are given by It is straightforward from (6.8) that ⋆ is commutative. t is the unit element of ⋆: using that t is the unit element over X, We obtain a • = a ↓ < a for a ∈ H from (6.10), therefore, ′ ⋆ in (6.9) is clearly an order reversing involution by (3.1).
As for the associativity of ⋆, first notice that (6.11) x ⋆ y ∈ H if and only if x, y ∈ H.
Indeed, if x, y ∈ H then x ⋆ y = h(x)h(y) / ∈ H, hence we are done. Next notice that (6.12) for Indeed, if the first row of (6.8) defines the value of x⋆y then x⋆y = (h(x)h(y)) ↓ and h(x)h(y) ∈ H, and hence h(x ⋆ y) = h((h(x)h(y)) ↓ ) whereas if x ⋆ y (6.8) = h(x)h(y) then h(x ⋆ y) = h(h(x)h(y)) = h(x)h(y) since h maps to X, X is closed under ·, and h is the identity on X. Now, h((x ⋆ y) ⋆ z) = h(x ⋆ (y ⋆ z)) readily follows from (6.12) and the associativity of ·. Therefore, by (6.7), (x ⋆ y) ⋆ z = x ⋆ (y ⋆ z) can only be possible if one side is in H and the other side is in H • . However, (6.11) shows that if one side is in H then x, y, z ∈ H, and hence, since H is closed under ⋆ by (6.11), also the other side must be in H.
It is easily seen that (6.13) ⋆ is isotone with respect to ≤ Y .
Indeed, let x, y, z ∈ Y with y < Y z. Since h and · are increasing with respect to . But then the only way for x⋆y ≤ Y x⋆z to hold is if x⋆y = h(x)h(y) and x⋆z = (h(x)h(z)) ↓ , which by (6.8) leads to assuming ¬(x, z ∈ H), h(x)h(z) ∈ H and either h(x)h(y) / ∈ H or x, y ∈ H. It follows that x, y ∈ H. But then h(x)h(y) = xy < xh(z) ≤ h(x)h(z) is a contradiction, where the strict inequality follows from y < h(z) and since x has an inverse.
To prove that ⋆ is residuated, and that x → ⋆ y = (x ⋆ y ′ ⋆ ) ′ ⋆ , by Lemma 3.3 it suffices to verify that t • is a dualizing element of (Y, ≤, ⋆). Since ′ ⋆ is clearly an order reversing involution, it suffices to verify that that x → ⋆ t ↓ exists and is equal to x ′ ⋆ . It amounts to verifying only three cases.
where in the last equality we used that H is a subgroup, hence x ′ is the inverse of x in X. On the other hand, for z > x ′ ⋆ it follows that z ≥ x ′ and hence x ⋆ z (6.13) ≥ x ⋆ x ′ x ′ ∈H, (6.8) = xx ′ = t > t ↓ . Therefore, by residuation, x → ⋆ t ↓ exists and is equal to We obtain x ⋆ x ′ ⋆ = y ↓ ⋆ y ′ (6.8) = (yy ′ ) ↓ = t ↓ . On the other hand, for z > x ′ ⋆ it follows that z ≥ x ′ and hence x ⋆ z (6.13) ≥ y ↓ ⋆ x ′ (6.8) = (yx ′ ) ↓ = (yy ′ ↑ ) ↓ . Here yy ′ ↑ > t holds by residuation since holds by (6.8) and by residuation, respectively. Summing up, the falsum-free reduct of Y is an involutive commutative residuated lattice with residual complement operation ′ ⋆ . By the second row of (6.9), t ′ ⋆ = (t ′ ) ↓ = t ↓ , and t ↓ is idempotent by the second row of (6.8). A particular instance of (6.10) shows that t is the unique cover of t • , hence t • = t ↓ and thus Y is even.
We are ready to prove that X = (X,∧,∨, ·, →, t, t) is an odd involutive FL e -algebra. (X,∧,∨) is a lattice. Indeed, all elements of H • are meet-irreducible because of claim (iv), hence X is closed under∧ (clearly,∧ is the restriction of ∧ to X). Commutativity of∨ is straightforward, and using claim (iv) a moment's reflection shows that∨ is associative, too, and the absorption law holds for∨ and∧. Commutativity of · is straightforward. X is closed under · since by claim (iv), (x ⋆ y) ↑ ∈ H if x ⋆ y ∈ H • . t ∈ X holds by claim (iii). Since t ∈ H, ⋆ (2,1) and ⋆ (2,2) show that for y ∈ X, t ⋆ y / ∈ H • holds, hence ty = t ⋆ y and thus t is the unit element for · over X since it is the unit element for ⋆ over Y . As for the associativity of ·, notice that (6.16) for Table 3 readily Hence, (X, ·) being the homomorphic image of a semigroup, is a semigroup. Next we prove that · is residuated. For x, y ∈ X, x → y = max{z ∈ X : xz ≤ y}. Here If x ⋆ z ∈ H • then since y ∈ X, (x ⋆ z) ↑ ≤ y holds if and only if x ⋆ z ≤ y holds by claim (iv). Therefore, max{z ∈ X : xz ≤ y} = max{z ∈ X : x ⋆ z ≤ y} holds yielding x → y = x → ⋆ y. Since x → y ∈ X, it also follows that x → y Involutivity of ′ t is seen as follows. We will verify that This, combined with (3.1) and that x ′ t ∈ H if and only if x ∈ H (shown by claims (vi) and (i)) concludes the proof of the statement. Clearly, Equality cannot hold since x ⋆ x ′ t = t by claim (i), whereas x ⋆ x ′ ⋆ ≤ f < t by residuation and since Y is even. Assume that there exits a ∈ X such that x ′ t > a > x ′ ⋆ . Then and hence x ⋆ f < a ′ ⋆ < x, a contradiction to claim (iv). Let x ∈ X \ H. If x ′ t > x ′ ⋆ then, as above, x ⋆ f < x follows, a contradiction to x / ∈ H. Finally, X is clearly odd, since the constant which defines the involution ′ t is the unit element.
Next we prove that H is a cancellative subalgebra of X. Indeed, H is closed under ⋆, shown by ⋆ (2,2) , t ∈ H holds by claim (iii), and H has an inverse operation ′ t , see claims (i) and (vi), hence H is cancellative by claim (xi) in Lemma 3.1. It is a subalgebra of X by claim (iii).
Finally, we verify that Y is the H-split of X. Indeed, the universe is as expected, see (6.15). The elements of H • are just below the respective elements of H, as they should be, see claim (iv). Finally we verify that ⋆ coincides with the product operation of the H-split of X.
: If x, y ∈ H then x ⋆ y ∈ H by ⋆ (2,2) . Hence x ⋆ y / ∈ H • by claim (ii), yielding x ⋆ y (6.3) = xy By the first row of Table 3, for any y ∈ Y , x ⋆ y = x ⋆ h Y (y). Since , as required in (6.8). : Assume ¬(x, y ∈ H) and h Y (x)h Y (y) ∈ H. By (6.16), h Y (x ⋆ y) ∈ H, hence x ⋆ y ∈ H ∪ H • . Since ¬(x, y ∈ H), by Table 3 it follows that x ⋆ y ∈ H • . Table 3 and x ⋆ y ∈ H • also implies that either x, y ∈ X \ H, or at least one of x and y is in H • and the other is in H ∪H • . In all these cases, by Table 3 and (6.2) it follows that ) ↓ , as required in (6.8).
(2b): Since h Y maps onto X, to prove that X is cancellative it suffices to prove that each element of h Y (Y) has inverse, which holds since for x ∈ Y , t∈H, (6.7) = t.
(2c): It is obvious that h Y preserves the unit element and the falsum constant. The definitions in (6.2)-(6.4) readily yield that h Y preserves the meet, the join, the product and the residual operation.

Bunches of layer algebras vs. bunches of layer groups
We introduce the notion of bunches of layer groups, and show that every bunch of layer algebras can be represented by a unique bunch of layer groups.
Definition 7.1. Let (κ, ≤ κ ) be a totally ordered set with least element t, and let an ordered triple κ I ,κ J , {t} be a partition of κ, whereκ I andκ J can also be empty. Define κ o , κ J , and κ I by one of the rows of Table 1, and let κ = κ o , κ J , κ I , ≤ κ .
Let G u = (G u , u , · u , −1u , u) be a family of abelian o-groups indexed by elements of κ, an let H u = (H u , u , · u , −1u , u) be a family of abelian o-groups indexed by elements of κ I , such that (7.1) for u ∈ κ J , G u is discrete, for u ∈ κ I , H u ≤ G u , and such that for u, v ∈ κ, u ≤ κ v, there exist homomorphisms ς u→v : G u → G v satisfying (G1) ς u→u = id Gu and ς v→w • ς u→v = ς u→w (direct system property), (G2) for v > κ u ∈ κ J , ς u→v (u) = ς u→v (u ↓u ), (G3) for u < κ v ∈ κ I , ς u→v maps into H v . Call G = G u , H u , ς u→v κ a bunch of layer groups.
Lemma 7.2. The following statements hold true.

3)
H u = (H u , u , · u , −1u , u) = λ(π 2 (X u )), κ = κ o ∪ κ J ∪ κ I , and for u, v ∈ κ such that u ≤ κ v, ς u→v : G u → G v is defined by Call G A the bunch of layer groups derived from A. (2) Given a bunch of layer groups G = G u , H u , ς u→v κ with κ = κ o , κ J , κ I , ≤ κ , A G = X u , ρ u→v κ is bunch of layer algebras, called the bunch of layer algebras derived from G, where where h u is the canonical homomorphism of X u . (3) Given a bunch of layer groups G, it holds true that G (AG ) = G, and given a bunch of layer algebras A, it holds true that A (GA) = A.
(2): X u defined in (7.5) is an involutive FL e -chain satisfying (4.1): if u ∈ κ o then see Lemma 3.2, if u ∈ κ J then see Lemmas 3.2 and 5.2, if u ∈ κ I then see Lemma 3.2 and Theorem 6.3. The ρ's defined in (7.6) are well defined since X u = G u holds by (7.5) if u / ∈ κ I , and if u ∈ κ I then h u maps to the universe of ι(G u ) (see (6.2)), which is G u . The ρ's are residuated lattice homomorphisms, since the (totally ordered group) homomorphisms ς u→v from G u to G v naturally extend to homomorphisms from the residuated lattice reduct of ι(G u ) to the residuated lattice reduct of ι(G v ) via claim (2) of Lemma 3.3, and hence ρ u→v can be regarded as the composition of residuated lattice homomorphisms (ς's and h's). The ρ's satisfy (A1): Notice that for u < v, since so does ς u→v . Over X u , ρ u→u Therefore, it suffices to prove the other condition in (A1) for u < v < w only: (A2): For u / ∈ κ o we have already seen that = ςu→v(u) = ςu→v(u ′ u ↑ ) (7.8) = ςu→v(u) (6.7) = (ςu→v • hu)(u) (7.6) = ρu→v(u) if u ∈ κ I .

The representation theorem
The main theorem of the paper is a representation theorem of odd or even involutive FL e -chains by bunches of layer groups. Lemmas 4.2 and 7.2 prove Theorem 8.1. The second part of Theorem 8.1 presents the direct constructional correspondence between odd or even involutive FL e -chains and bunches of layer-groups, that is, one without referring to the intermediate explanatory step of layer algebras. Theorem 8.1. For every odd or even involutive FL e -chain X there exists a unique bunch of layer groups G such that X is the involutive FL e -chain derived from the bunch of layer algebras derived from G, in notation, X = X AG . Conversely, for every bunch of layer groups G there exists a unique odd or even involutive FL echain X such that G is the bunch of layer groups derived from the bunch of layer algebras of X, in notation, G = G AX . In more details: (A) Given an odd or an even involutive FL e -chain X = (X, ≤, ·, →, t, f ) with residual complement operation ′ , G X = G u , H u , ς u→v κo,κJ ,κI ,≤κ is bunch of layer groups, called the bunch of layer groups of X, where κ = {x → x : x ∈ X} = {u ≥ t : u is idempotent} is ordered by ≤, κ I = {u ∈ κ \ {t} : u ′ is idempotent}, κ J = {u ∈ κ \ {t} : u ′ is not idempotent}, κ o , κ J , κ I are defined by Table 4,  Table 4. for x, y ∈ X u , if u ∈ κ I , h u (x) · u h u (y) / ∈ H u or x, y ∈ H u x · u y if u / ∈ κ I , for x ∈ X u and y ∈ X v , xy = ρ uv (x) • uv ρ uv (y), for x ∈ X, , for x, y ∈ X, x → y = (xy ′ ) ′ , → is the residual operation of ·, t is the least element of κ, f is the residual complement of t, and is given by In addition, ρ v (x) = vx for v ∈ κ and x ∈ X, X X is odd if t ∈ κ o , even with a non-idempotent falsum if t ∈ κ J , and even with an idempotent falsum if t ∈ κ I . (C) Items (A) and (B) describe a one-to-one correspondence between the class containing all odd and all even involutive FL e -chains and the class of bunches of layer groups: given a bunch of layer groups X it holds true that X (XX ) = X , and given an odd or even involutive FL e -chain X it holds true that X (XX) ≃ X 19 .
Example 8.2. We present the bunch representation of a few known structures, among which are the two extremal classes (abelian o-groups and odd Sugihara chains) mentioned in the introduction. Denote by ½ the trivial (one-element) group.
19 If Definition 7.1 is slightly modified in such a way that the • Hu's are "stored" in the definition of a bunch (like X = Gu, Hu, • Hu, ςu→v κo,κ J ,κ I ,≤κ ), and instead of taking a copy • Hu of Hu, that stored copy is used in the construction of Theorem 8.1/(A), then also X (X X ) = X holds. Then, Theorem 8.1 describes a bijection, in a constructive manner, between the classes of odd or even involutive FLe-chains and the class of bunches of layer groups.
• Finite partial sublex products of abelian o-groups have been shown in [32,30] to be exactly those odd involutive FL e -chains which have finitely many positive idempotent elements. These are exactly the algebras X G , where κ is finite in G = G u , H u , ς u→v {t},κJ ,κI ,≤κ . • Algebras which can be constructed by the involutive ordinal sum construction of [29] are exactly the algebras X G , where G = G u , ½ u , ς u→v {t},∅,κ\{t},≤κ .
By (G3), also the homomorphisms are trivial. • Algebras which can be constructed by the consecutive application of the ordinal sum construction as defined in [18] are exactly the algebras X G , where κ is finite in G = G u , ½ u , ς u→v 0 {t},∅,κ\{t},≤κ . By (G3), also the homomorphisms are trivial.
Remark 8.3. If X is densely ordered in Theorem 8.1 then the H u 's in the representation G = G u , H u , ς u→v κ of X are uniquely determined by the rest of G. Therefore, if X is densely ordered then the representation of X by layer groups can be written in a simpler form of G u , ς u→v κo,κJ ,κI ,≤κ . To prove it we state that for u ∈ κ I , H u = κ∋s<κu ς s→u (G s ).
Remark 8.4. The easiest way of generalizing Theorem 8.1 to conic algebras is to observe that there exist no conic odd or even involutive FL e -algebras which are not linearly ordered. Indeed, let a, b ∈ X. Proving that a and b are comparable, that is, a ≤ b or b ≤ a amounts to proving at ≤ b or bt ≤ a, or equivalently, a → b ≥ t or b → a ≥ t by adjointness. If a → b ≥ t then a → b < t since X is conic, hence a → b ≤ f since X is odd or even. By claims (i) and (ii) in Lemma 3.1, ab ′ ≥ t follows. Therefore, (a ′ b) ′ ≥ t holds by claim (iii) in Lemma 3.1, hence b → a ≥ t holds by claim (ii) in Lemma 3.1.