L-algebras arise in algebraic logic, in the theory of one-sided lattice-ordered groups, and in connection with set-theoretic solutions of the quantum Yang–Baxter equation. They apply in several ways to Garside groups. For example, the set of primitive elements, the set of simple elements, and the negative cone of a Garside group are all L-algebras. Picantin’s iterated crossed product decomposition of Garside groups can be reformulated and extended in terms of L-algebras. It is proved that the structure group of an L-algebra, introduced in connection with the “logic” of -groups, is torsion-free. This applies to the left group of fractions of not necessarily noetherian, Garside-like monoids which need not embed into their ambient group.
A topological proof that Artin’s braid groups are torsion-free was given in 1962 by Fadell, Fox, and Neuwirth [12, 13]. The first algebraic proof made use of the orderability of braid groups [5, 16]. Later, Dehornoy  showed that all Garside groups are torsion-free, and gave a much simpler proof  without the noetherian hypothesis. The method of simplifying proofs by dropping redundant hypotheses came to a certain end with a one-line proof that every group with a right invariant lattice order is torsion-free (see [20, Proposition 3]). On the other hand, Dehornoy’s slightly longer proof  applies to an even more general situation. In the terminology of Section 1, his result shows that the group of fractions of a cancellative left hoop (a residuated monoid) is torsion-free.
In this paper, we prove that left cancellability can be dropped. More generally, we replace the left hoop by a unital cycloid , a set with a single binary operation which satisfies
The relevance of equation (0.1) will be explained in Section 1. We give an example where the natural map from the unital cycloid to its structure group – the “group of fractions” – is surjective. So our theorem does not rely on any kind of partial ordering of the group.
1 The structure group of an L-algebra
Equation (0.1) exhibits a close relationship between group theory, algebraic logic, and quantum structures . It first occurred in a logical context [2, 21]. Here stands for the logical implication (“x implies y”). If the left multiplications are bijective, such a set is called a cycle set . By [17, Proposition 1], cycle sets are in one-to-one correspondence with non-degenerate unitary set-theoretic solutions of the quantum Yang–Baxter equation. Thirdly, equation (0.1) occurs in the theory of lattice-ordered groups ( -groups for short) [1, 4], the negative cone of an -group G being a cycloid with respect to
In the latter two examples, there is an associative multiplication which should be distinguished from the operation . Therefore, we write instead of , which reflects the logical origin of that operation. In the presence of a multiplication, we then have
By this equivalence, the unit element of an -group G satisfies
for all . A cycloid with such an element 1 (a logical unit ) is said to be unital. (In propositional logic, 1 stands for the “true” proposition.)
For a unital cycloid , the relation
is a congruence. If this relation is trivial, that is, implies , then X is called an L-algebra . Thus every unital cycloid X gives rise to an L-algebra . Every L-algebra has a partial order
similar to the logical entailment relation.
The negative cone, and every interval of an -group is an L-algebra in two ways, namely, with respect to each of the operations and . In several respects, L-algebras play a fundamental part in the theory of Garside groups. For example, the primitive elements as well as the simple elements [10, 7] of a Garside group form an L-algebra. Picantin’s iterated crossed products of Garside groups  admit a simple formulation in terms of L-algebras . More generally, L-algebras arise in connection with right -groups, that is, groups with a right invariant partial order. For example, the structure groups of non-degenerate unitary set-theoretic solutions of the quantum Yang–Baxter equation are right -groups with a distributive lattice structure . For finite X, these groups are Garside [3, 9].
In  we associate a structure group to any L-algebra X. To explain this, we have to clarify the rôle of the product in (1.1). Define a left hoop  to be a monoid H with a binary operation satisfying
for . By [18, Propositions 3 and 4], every left hoop is an L-algebra, and any pair has a meet
Therefore, the partial order of H is given by
A left hoop H is self-similar if it is right cancellative. By [18, Proposition 5], this can be expressed in terms of equations:
For a left hoop H, the following are equivalent.
H is self-similar, that is, holds for .
for all .
for all .
Since follows by (b), a self-similar left hoop is given by the axioms
Note that by equation (1.1), self-similarity is a property of L-algebras. In fact, an L-algebra X is self-similar if and only if for any , the map gives a bijection (see [18, Definition 2]). Moreover, self-similarity implies that X admits a unique multiplication which makes X into a left hoop.
Now let H be a self-similar L-algebra with an L-subalgebraX, which means that the operation of X is induced by that of H. As remarked above, H can be regarded as a left hoop. If the monoid H is generated by X, we call H a self-similar closure of X. By [18, Theorem 3], a self-similar closure of any L-algebra X exists and is unique, up to isomorphism. We denote it by . In particular, every self-similar left hoop H satisfies
Since any self-similar left hoop satisfies the left Ore condition, the self-similar closure of an L-algebra X admits a left group of fractions with a natural map
Classical logic is built upon the two-element L-algebra , which is completely determined by the property of 1 being a logical unit. Thus , where a and 1 stand for the two possible truth values. The self-similar closure is the negative cone of the -group , the additive group of integers. Thus consists of the powers with for . As the two-element L-algebra generates the group , this shows that logic creates arithmetic, the theory of the ring .
If the pair of truth values is replaced by the interval , we obtain an L-algebra with , which is fundamental for measure and integration theory . Its self-similar closure is the negative cone of , the additive group of the reals.
Let V be an -dimensional Euclidean space, and let be the corresponding elliptic real n-space, that is, the projective space of one-dimensional subspaces of V. The scalar product of V defines an elliptic polarity which associates a hyperplane to any point . “Elliptic” means that for all . Let denote the lattice of -linear subspaces of V. Thus gives a lattice isomorphism between the and the projective subspaces of . Define
for . Then is a logical unit, and
Furthermore, , which yields
Note that . For , this implies that
which shows that is an L-algebra. Assume that H and L are orthogonal ( ) in the sense that or equivalently, . Then
So , which yields
Thus coincides with the product whenever . Let be the L-subalgebra of subspaces of codimension . Since every is a finite meet of pairwise orthogonal hyperplanes, it follows that . By [20, Proposition 19], is a modular lattice.
Furthermore, it can be shown that the equation
holds for distinct . By [20, Theorem 4], this implies that is a Garside group. The example gives an L-algebraic interpretation of the well-known Gram–Schmidt process in the Euclidean space .
For a unital cycloid X, we define the structure group to be . Note that in general, the map (1.5) need not be injective. Indeed, two elements of an L-algebra X satisfy if and only if holds for some . Thus q is injective if and only if is left cancellative. Two elements with are called equipollent . If X has a smallest element 0 ( “false”), equipollence of x and y is equivalent to . Our aim is to show that the structure group of any L-algebra X is torsion-free. Thus, by (1.5), we can assume, without loss of generality, that X is a self-similar left hoop.
Let H be a self-similar left hoop. For a pair of elements , we define the derived pair by
The n-th derived pair will be denoted by for . In particular, and . The notation does not express the dependency of on both a and b, which should always be clear from the context. Furthermore, we write
for . Thus . More generally, we abbreviate
for in . The concept of derived pair (1.6) is justified by
Let H be a self-similar left hoop. Two elements are equipollent if and only if and are equipollent.
By Proposition 1, we have
So is equivalent to the conjunction of and . Thus a and b are equipollent if and only if for some . By [18, Proposition 12], the latter condition is equivalent to .∎
The product expressions (1.7) satisfy the following.
Let H be a self-similar left hoop. For and ,
We proceed by induction over the sum . For or , the equations are obvious. Assume that the equations hold for a fixed sum . For with , using Proposition 1, this gives
By symmetry, the lemma is proved. ∎
The expressions in equation (1.8) admit the following interpretation.
Let H be a self-similar left hoop. For and ,
We prove the first equation by induction on n. For , the equation is trivial. Thus, assume that the equation holds for a fixed n. Then Lemma 1 gives
which completes the inductive step.
The second equation is proved analogously. ∎
The following corollary is essentially due to Dehornoy .
Let H be a self-similar left hoop. For and ,
This follows immediately by equation (1.3). ∎
Let H be a self-similar left hoop. For and , we have
Equation (1.11) can be rewritten as
For , this is obvious. Assume the equation holds for some n. Then equation (1.10) gives
So the corollary follows by induction. ∎
Let H be a self-similar left hoop. For a pair , we define a weak -cycle to be a non-empty sequence with
If , we speak of an -cycle.
An -cycle can be visualized by a periodic diagram, e. g., for :
So we can assume that is an n-periodic sequence defined for all , and there is a single condition for all .
Let be a self-similar left hoop. If the sequence is an -cycle, then is an -cycle.
For , equation (1.10) yields . Furthermore, , which proves the claim. ∎
If a self-similar left hoop H admits an -cycle, then .
Let be an -cycle. We show that
which proves equation (1.12). In particular, . Hence
which yields . Similarly, implies that . By Proposition 2, this proves that a and b are equipollent. ∎
If a self-similar left hoop H admits a weak -cycle, then is equal to .
Let be a weak -cycle. So there is an element with . Choose for all . Since , there exist elements with for all . In particular, shows that . Hence we have . Furthermore, gives for , and yields . So the extend to an n-periodic sequence with and
for all . Hence
for all . Similarly,
which shows that the element
does not depend on i. Similarly,
So the element
does not depend on i. Hence we obtain
for all . Thus is a -cycle. By Lemma 2, this shows that p and q are equipollent. In particular, and are equipollent. So Proposition 2 implies that and are equipollent. Since , there is an element with . Hence we have and . Thus and are equipollent. Again by Proposition 2, this yields . ∎
Now we are ready to prove our main result.
The structure group of an L-algebra X is torsion-free.
Every element of is of the form with . Assume that for some . Then equation (1.11) gives
Define for . Then equation (1.10) gives
for . Furthermore,
Since every self-similar left hoop H is an L-algebra and satisfies , we have:
The left group of fractions of a self-similar left hoop H is torsion-free.
2 Examples and constructions
We start with two new constructions of self-similar left hoops. Let M be a self-similar left hoop. By we denote the group of bijections
with for all . The equivalence (1.4) shows that consists of order automorphisms, that is,
holds for and . Note that the equation is valid also for , while for , it turns into . So the can be conceived as right multiplications with an “external” element .
Let H and M be self-similar left hoops, and let be a monoid homomorphism. We write . Define the lexicographic semidirect product to be the set of all formal products ax with and such that
for and .
Let H be a monoid with a partial order (1.4) and a binary operation which satisfies for all . Then H is a left hoop.
The inequality is trivial. For all , we have
Hence we have . To verify equation (1.3), we note first that . Conversely, assume that . Then for some . Hence , and thus for some . So we obtain . ∎
Let H and M be left hoops with M self-similar. The lexicographic semidirect product is a left hoop which satisfies
Moreover, is self-similar if and only if H is self-similar.
We show first that the multiplication (2.1) is associative. So we have to verify
for and . For , this follows by the associativity of H. Assume that . Since , the left-hand side of equation (2.3) becomes . If , the right-hand side is . So the equation follows since . Similarly, for , equation (2.3) follows by . Furthermore, the multiplication (2.1) admits a neutral element. Thus is a monoid.
In what follows, let denote the inverse of the action , that is,
for and . Now we define a second operation on by
Next we verify the equivalence
in . Assume that . Then the left-hand side states that , that is, or ( and ). For , both sides of (2.5) are true, while for , both sides of (2.5) are equivalent to . So we can assume that . The left-hand side of (2.5) is then equivalent to , that is,
If y and z are incomparable, . So the right-hand side of (2.5) becomes , that is, . Since , this is equivalent to . For , the right-hand side of (2.5) is , which gives either or ( and ). Since M is self-similar, is equivalent to . So the right-hand side of (2.5) can be transformed into (2.6). Finally, since , both sides of (2.5) are true if . This completes the proof of (2.5).
holds in . For , we have . So equation (2.7) becomes . If , then
To any totally ordered abelian group G we associate a self-similar left hoop as follows. Let be the free monoid over G. To distinguish the multiplication of with that of G, we write the elements of as . As G is an additive group, there is no confusion between the neutral element 0 of G and the unit element of , the empty word 1. We define to be the free monoid over G with the relation
Let G be a totally ordered abelian group. Then is a self-similar left hoop.
Using equation (2.8), the elements of can be put into the normal form with . The existence of such a form is trivial. Uniqueness follows by considering the maximal intervals with for all . Thus, from now on, we represent the elements of by their normal form. Multiplying from the left by some , we get . If , multiplying by for some yields . So the natural ordering (1.4) provides with a linear, lexicographic order. Note, however, that the map with
is neither increasing nor decreasing.
For and , it is easily checked that implies . Hence is right cancellative. It follows that, if , there is a unique with . If , we set . Hence we have . In particular, holds for all . Since , it follows that , that is, we have . Hence . Thus
which yields . ∎
Let be the L-algebra of Example 1. Consider the monoid H generated by with a single relation . It is easily checked that the elements of H can be put into the normal form
with . The elements with form a submonoid M. To any , we associate the sequence if , and if . Then a straightforward calculation shows that M is isomorphic to , and
a self-similar left hoop with structure group . The canonical map is given by
In general, the image of the map is a submonoid of , but need not be trivial. So the partial order of H need not induce a partial order of . In the present example, .
The author thanks the referee for helpful remarks which led to an improvement of the paper.
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