Deciding Equations in the Time Warp Algebra

Join-preserving maps on the discrete time scale $\omega^+$, referred to as time warps, have been proposed as graded modalities that can be used to quantify the growth of information in the course of program execution. The set of time warps forms a simple distributive involutive residuated lattice -- called the time warp algebra -- that is equipped with residual operations relevant to potential applications. In this paper, we show that although the time warp algebra generates a variety that lacks the finite model property, it nevertheless has a decidable equational theory. We also describe an implementation of a procedure for deciding equations in this algebra, written in the OCaml programming language, that makes use of the Z3 theorem prover.


Introduction
Graded modalities [FKM16, GKO + 16] provide a unified setting for describing effects of a program, such as which parts of memory it modifies [LG88], or the resources it consumes, such as how long it takes to run [GS14].Given a type A and grading f , the new type □ f A represents a modification of A that incorporates the behavior prescribed by f .The gradings themselves are often equipped with an ordered algebraic structure that is relevant for programming applications.Typically, they form a monoid, relating □ gf A to □ f □ g A, or admit a precision order along which the graded modality acts contravariantly, i.e., for f ≤ g, there exists a generic program of type □ g A → □ f A that allows movement from more to joins of basic terms, constructed using just the monoid operations and involution.It follows that the equational theory of W is decidable if there exists an algorithm that decides W |= e ≤ t 1 ∨ • • • ∨ t n for arbitrary basic terms t 1 , . . ., t n , where e is interpreted as the identity map.We provide such an algorithm by relating the existence of a counterexample to e ≤ t 1 ∨ • • • ∨ t n to the satisfiability of a corresponding first-order formula in ω + , understood as an ordered structure with a decidable first-order theory, and describe an implementation, written in the OCaml programming language, that makes use of the Z3 theorem prover.
Overview of the paper.In Section 2, we introduce the time warp algebra W as an example of an involutive residuated lattice consisting of join-preserving maps on a complete chain (totally ordered set), and establish some of its elementary properties.In particular, we show that W is simple, has no finite subalgebras, and generates a variety (equational class) that lacks the finite model property.We also describe a subalgebra R of W consisting of time warps that are 'regular' in the sense that they are eventually either constant or linear, and provide an explicit description of the involution operation.
In Section 3, we describe the existence of a potential counterexample to e ≤ t 1 ∨ • • • ∨ t n , where t 1 , . . ., t n are basic terms, using the notion of a diagram, which provides a finite partial description of an evaluation of terms as time warps.As a consequence, an equation is satisfied by W if, and only if, it is satisfied by R (Theorem 3.11).Finally, we use the resulting description to reduce the existence of a counterexample to the satisfiability of a first-order formula in ω + , understood as an ordered structure with a decidable first-order theory, thereby establishing the decidability of the equational theory of W (Theorem 3.12).
In Section 4, we describe an implementation of our decision procedure for the equational theory of W, written in the OCaml programming language [LDF + 22], that makes use of the Z3 theorem prover [dMB08].Finally, in Section 5, we consider some potential avenues for further research; in particular, we explain how to adapt the decision procedure to deal with extra constants for first-order definable time warps such as ⊥ and ⊤, and explore a relational approach to the study of the time warp algebra and related structures.

The Time Warp Algebra
In this section, we introduce and establish some elementary properties of the time warp algebra W in the general framework of involutive residuated lattices.In particular, we show that the variety generated by W lacks the finite model property and that 'regular' time warps-those that are eventually either constant or linear-form a subalgebra of W. We also provide an explicit description of the involution operation on time warps.
A pointed residuated lattice (also known as an FL-algebra) is an algebraic structure L = ⟨L, ∧, ∨, •, \, /, e, f⟩ of signature ⟨2, 2, 2, 2, 2, 0, 0⟩ such that ⟨L, •, e⟩ is a monoid, ⟨L, ∧, ∨⟩ is a lattice with an order defined by a Moreover, it follows from the existence of residual operations \ and / that the operation • preserves existing joins in both coordinates; in particular, for all The structure L is said to be distributive if its lattice reduct ⟨L, ∧, ∨⟩ is distributive and fully distributive if it is distributive and, for all The element f ∈ L is called cyclic if f/a = a\f, for all a ∈ L, and dualizing if, for all a ∈ L, If f is both cyclic and dualizing, then the map The class of pointed residuated lattices such that f is cyclic and dualizing forms a variety (equational class) that is term-equivalent to the variety of involutive residuated lattices: algebraic structures ⟨L, ∧, ∨, •, ′ , e⟩ of signature ⟨2, 2, 2, 1, 0⟩ such that ⟨L, •, e⟩ is a monoid, ⟨L, ∧, ∨⟩ is a lattice, ′ is an involution on ⟨L, ∧, ∨⟩, and for all a, b, c The term-equivalence is implemented by defining x ′ := x\f and, conversely, x\y := (y ′ x) ′ , y/x := (xy ′ ) ′ , and f := e ′ (see [MPT23] for further details).For any element a of an involutive residuated lattice, we also define inductively a 0 := e and a k+1 := a k • a (k ∈ N).Now let Tm denote the term algebra of the language of involutive residuated lattices defined over a countably infinite set of variables Var, and call a term t ∈ Tm basic if it is constructed using the operation symbols •, ′ , and e.As usual, an equation is an ordered pair of terms s, t ∈ Tm, denoted by s ≈ t, and s ≤ t abbreviates s ∧ t ≈ s.An involutive residuated lattice L satisfies an equation s ≈ t, denoted by L |= s ≈ t, if s h = t h for every homomorphism h : Tm → L, where u h := h(u) for u ∈ Tm.
In a fully distributive involutive residuated lattice, every term is equivalent to a meet of joins of basic terms.More precisely: Lemma 2.1.There exists an algorithm that produces for any term t ∈ Tm positive integers m, n 1 , . . ., n m and basic terms t i,j for each i ∈ {1, . . ., m} and j ∈ {1, . . ., n i } such that for any fully distributive involutive residuated lattice L, Proof.The desired basic terms are obtained by iteratively distributing joins over meets and the monoid multiplication over both meets and joins, and pushing the involution inwards using the De Morgan laws.
Next, let C be any complete chain with a least element 0 and a greatest element ∞.The set Res(C) of maps on C that preserve arbitrary joins forms a fully distributive pointed residuated lattice Res(C) = ⟨Res(C), ∧, ∨, •, \, /, id, p⟩, where ∧, ∨ are defined pointwise, • is functional composition, id is the identity map, and p is the join-preserving map The least and greatest elements of Res(C) are the maps ⊥, ⊤ satisfying, respectively, ⊥(x) = 0, for all x ∈ C, and ⊤(0) = 0 and ⊤(x) = ∞ for all x ∈ C\{0}.Moreover, it follows from [EGGHK18, Proposition 2.6.18] and [San20, Section 4]) that p is the unique cyclic and dualizing element of Res(C) and hence that Res(C) is term-equivalent to the involutive residuated lattice ⟨Res(C), ∧, ∨, •, ⋆ , id⟩, where f ⋆ := f \p = p/f for each f ∈ Res(C).
Recall that an algebraic structure A is simple if Con(A) = {∆ A , ∇ A }, where Con(A) denotes the set of congruences of A, ∆ A = {⟨a, a⟩ | a ∈ A}, and ∇ A = A × A.
Proposition 2.2.For any complete chain C, the pointed residuated lattice Res(C) is simple.
Proof.Consider any Θ ∈ Con(Res(C))\{∆ Res(C) }.Since Θ ̸ = ∆ Res(C) and a congruence of a pointed residuated lattice is fully determined by the congruence class of its multiplicative unit (see [BT03]), there exists an f ∈ Res(C)\{id} such that f Θ id.Choose any c ∈ C such that f (c) ̸ = c.We consider the cases f (c) < c and c < f (c).Suppose first that f (c) < c and define the map g ∈ Res(C) such that for each d ∈ C, Since f Θ id and Θ is a congruence, also ((g But Θ is a lattice congruence, so its congruence classes are convex and Θ = ∇ Res(C) .Suppose next that c < f (c).Then and, since f is order-preserving and C is a chain, (f \id)(c) < c.But then, since f \id Θ id\id = id, as in the previous case, Θ = ∇ Res(C) .
Let us focus our attention now on the special case of the successor ordinal ω + = ω ∪ {ω}.For convenience, we call both the pointed residuated lattice Res(ω + ) and the corresponding term-equivalent involutive residuated lattice, the time warp algebra W, referring to members of W , i.e., join-preserving maps on ω + , as time warps.Clearly, a map f : ω + → ω + is a time warp if, and only if, it is order-preserving and satisfies f (0) = 0 and f As motivation for subsequent sections, we will show below that equational reasoning in W cannot be checked by considering finite members of the variety generated by this algebra.
Observe first that p := id ⋆ ∈ W is the 'predecessor' time warp satisfying for m ∈ ω + , Clearly, p < id, so W satisfies the equation e ′ ≤ e.
The involution operation can be described using p as follows.
Lemma 2.3.For any time warp f and m ∈ ω + , Proof.Let h be the function defined by h(m) Clearly, h is order-preserving and satisfies h(0) = 0. Hence, to show that h is a time warp, it remains to check that h Observe next that p k+1 < p k for each k ∈ N and hence that the ∅-generated subalgebra of W is infinite.So W has no finite subalgebras.Moreover, W is simple, by Proposition 2.2, so any quotient of W is either trivial or isomorphic to W. Indeed, we do not know if the variety generated by W contains any non-trivial finite algebra, but can show at least that it cannot contain Res(C) for any finite chain C and does not have the finite model property. 1o this end, observe first that the element p has a right inverse; that is, p • s = id, where s is the 'successor' time warp satisfying for each m ∈ ω + , Hence p • (p\id) = id, and W satisfies the equation e ≈ e ′ • (e ′ \e).
Proposition 2.4.The variety generated by W does not have the finite model property and does not contain Res(C) for any finite chain C.
Proof.Let ⟨L, ∧, ∨, •, ′ , e⟩ be any finite member of the variety generated by W.Then, since e ′ ≤ e and L is finite, there exists a k ∈ N such that (e ′ ) k = (e ′ ) k+1 .Moreover, e ′ = e ′ e = (e ′ ) 2 • (e ′ \e), since e = e ′ • (e ′ \e) and e is the multiplicative unit of L, and hence, iterating this step, e ′ = (e ′ ) k+1 (e ′ \e) k = (e ′ ) k (e ′ \e) k = e.The equation e ′ ≈ e is therefore satisfied by all the finite members of the variety generated by W, but not by W itself, since p < id.Moreover, since Res(C) is finite for any finite chain C and does not satisfy e ′ ≈ e, it cannot belong to the variety generated by W.
Although the variety generated by W does not have the finite model property, we will show in this paper that it is generated by a subalgebra of W consisting of time warps that have a simple finite description.Let us call a time warp f ∈ W eventually constant if there exists an m ∈ ω such that f (n) = f (m) for all n ∈ ω with n ≥ m, eventually linear if there exist an m ∈ ω and a k ∈ Z such that f (n) = n + k for all n ∈ ω with n ≥ m, and regular if it is eventually constant or eventually linear.Equivalently, a time warp f ∈ W is regular if, and only if, there exist m ∈ ω, l ∈ {0, 1}, and k ∈ Z ∪ {ω} such that f (n) = ln + k for all n ∈ ω with n ≥ m.
Proposition 2.5.The set of regular time warps forms a subalgebra R of W.
Proof.Clearly, id is eventually linear, and hence regular.Suppose that f, g ∈ W are regular.It is easy to see that f ∧ g and f ∨ g are then also regular.If g is eventually constant, then there exists an m ∈ ω such that g(n) = g(m) for all n ∈ ω with n ≥ m, and hence also f (g(n)) = f (g(m)) for all n ∈ ω with n ≥ m, so f • g is eventually constant.Suppose then that g is eventually linear, that is, there exist an m ∈ ω and a k ∈ Z such that g(n) = n + k for all n ∈ ω with n ≥ m.If f is eventually constant, then there exists an l ∈ ω such that f (n) = f (l) for all n ∈ ω with n ≥ l, and hence f (g(n)) = f (l) for all n ∈ ω with n ≥ max(m, l − k), that is, f • g is eventually constant.If f is eventually linear, then there exist an l ∈ ω and a r ∈ Z such that f (n) = n + r for all n ∈ ω with n ≥ l, and hence It remains to show that f ⋆ is regular for f regular.Suppose first that f is eventually constant, that is, there exists an m ∈ ω such that f i.e., f ⋆ is eventually constant.Finally, suppose that f is eventually linear, that is, there exists an m ∈ ω and a k ∈ Z such that f (n) = n + k for all n ∈ ω with n ≥ m.Then every l ∈ ω with l ≥ m + k + 1 lies in the image of f and n = l − k is the unique solution for the equation l = f (n), so f ⋆ (l) = l − (k + 1), by Lemma 2.3, i.e., f ⋆ is eventually linear.
We conclude this section by providing an explicit description of the involution operation on time warps that will play a crucial role in the decidability proof in the next section.For any time warp f ∈ W , let last(f ) denote the smallest m ∈ ω + such that f takes the same value on m (and hence all elements greater than m) as ω.More formally: Observe that last(f ) < ω if, and only if, f is eventually constant.Moreover, we have last The behaviour of this operation with respect to compositions and involutions of time warps is easily described as follows.
Lemma 2.6.For any time warps f and g, Proof.For the first equivalence, observe that last(f g) = ω if, and only if, f g(m) < f g(ω) for all m ∈ ω.However, f g(m) = f g(ω) for some m ∈ ω if, and only if, either g(m) = g(ω) for some m ∈ ω, or g(m) < g(ω) for all m ∈ ω and f (k) = f (ω) for some k ∈ ω, which is equivalent to last(g) < ω or last(f ) < ω.For the second equivalence, observe that last(f ⋆ ) = ω if, and only if, f ⋆ (m) < f ⋆ (ω) for all m ∈ ω.But f ⋆ (m) = f ⋆ (ω) for some m ∈ ω if, and only if, f (m) = f (ω) for some m ∈ ω, by Lemma 2.3, which is equivalent to last(f ) < ω.
Finally, we are able to provide the promised explicit description of the involution operation on time warps.Proposition 2.7.For any time warp f , n ∈ ω\{0}, and m ∈ ω, Proof.Consider any n ∈ ω + \{0} and m ∈ ω.For the first two equivalences, observe that, using Lemma 2.3 and the assumption that n ̸ = 0, For the second two equivalences, observe first that, using Lemma 2.3,

Decidability via Diagrams
In this section, we turn our attention to the problem of deciding equations in the time warp algebra W. Observe first that for any equation s ≈ t with s, t ∈ Tm, Moreover, by Lemma 2.1, each u ∈ Tm is equivalent in W to a meet of joins of basic terms.Hence, since W |= e ≤ u 1 ∧ • • • ∧ u m if, and only if, W |= e ≤ u i , for each i ∈ {1, . . ., m}, we may restrict our attention to deciding equations of the form e ≤ t 1 ∨ • • • ∨ t n , where {t 1 , . . ., t n } is any finite non-empty set of basic terms.More precisely: Proposition 3.1.The equational theory of W is decidable if, and only if, there exists an algorithm that decides W |= e ≤ t 1 ∨ • • • ∨ t n for any basic terms t 1 , . . ., t n .
To address this problem, we relate the existence of a counterexample to W |= e ≤ t 1 ∨ • • • ∨ t n , where t 1 , . . ., t n are basic terms, to the satisfiability of a corresponding first-order formula in ω + , understood as an ordered structure with a decidable first-order theory.Such a counterexample is given by assigning time warps to the variables in t 1 , . . ., t n to obtain time warps t1 , . . ., tn satisfying id ̸ ≤ t1 ∨ • • • ∨ tn , that is, by a homomorphism h : Tm → W and element k ∈ ω + satisfying k > t i h (k), for each i ∈ {1, . . ., n}.The translation into a first-order formula is obtained in two steps.First, a 'time variable' κ is introduced to stand for the unknown k ∈ ω + and finitely many 'samples' are generated that correspond to other members of ω + used in the computation of t 1 h (k), . . ., t n h (k).Second, finitely many quantifier-free formulas are defined that describe the relationships between samples according to the semantics of W. The required formula is then the existential closure of the conjunction of these quantifier-free formulas and a formula that expresses κ > t i h (κ) for each i ∈ {1, . . ., n}.
Let us fix a countably infinite set T V of time variables, denoted by the (possibly indexed) symbol κ.We call any member of the language generated by the following grammar a sample, where t ∈ Tm is any basic term and κ ∈ T V is any time variable:

Now let ⇝ be the binary relation defined on the set of all samples by
and let ⇝ * denote the reflexive transitive closure of this relation.We call a sample set (i.e., set of samples) ∆ saturated if whenever α ∈ ∆ and α ⇝ β, also β ∈ ∆, and define the saturation of a sample set ∆ (analogously to the Fischer-Ladner closure of formulas in Propositional Dynamic Logic [FL79]) as Crucially, this definition yields the following property: Lemma 3.2.The saturation of a finite sample set is finite.
Running example part 1.Let us consider the equation e ≤ xx ′ as a running example throughout this section.In this case, the sample set of interest is {xx ′ [κ]}.In Figure 1, its saturation {xx ′ [κ]} ⇝ is visualized as a tree, where each parent node is related to its successors by ⇝ and redundant samples are omitted.
Consider next a first-order signature τ = {⪯, S, 0, ω} with a binary relation symbol ⪯, a unary function symbol S, and two constants 0 and ω.We denote by A the τ -structure with universe ω + and natural order ⪯ A , defining S A (n) := n + 1, for each n ∈ ω, S A (ω) := ω, ω A := ω, and 0 A := 0. Since no other τ -structure will be considered in this section, we will omit the superscripts from now on.We use the symbols ¬, ⋏, ⋎, ⇒, and ⇔ to denote the logical connectives 'not', 'and', 'or', 'implies', and 'if, and only if', respectively, and let Let us fix now a saturated sample set ∆ and consider its members as first-order variables.We define the following sets of first-order quantifier-free τ -formulas over ∆: The term 'diagram' recalls a similar concept used to prove the decidability of the equational theory of lattice-ordered groups in [HM79].
Running example part 4. Consider again ∆ = {xx ′ [κ]} ⇝ and the ∆-diagram δ : ∆ → ω + from parts 2 and 3 of the running example.The construction in the above proof yields the strong extension of ⌊x⌋ δ depicted in Figure 2, where the added relations are indicated with dashed arrows.Indeed, the obtained extension is the time warp id, which we already know to be a suitable extension.Lemma 3.5.Let t 1 and t 2 be basic terms.If f 1 and f 2 strongly extend ⌊t 1 ⌋ δ and ⌊t 2 ⌋ δ , respectively, then f 1 f 2 strongly extends ⌊t 1 t 2 ⌋ δ .

Concluding Remarks
The main contribution of this paper is a proof that the equational theory of the time warp algebra W is decidable, supported by an implementation of a corresponding decision procedure.There remain, however, many open problems regarding computational and structural properties of W and related algebras of join-preserving maps on complete chains.Below, we briefly discuss some of these problems and potential avenues for further research.
An axiomatic description.Although we have provided an algorithm to decide equations in the time warp algebra, an axiomatic description of its equational theory is still lacking.The algebra generates a variety of fully distributive involutive residuated lattices satisfying the equations e ′ ≤ e and e ≈ e ′ • (e ′ \e), but not much more is known.An axiomatization could also provide the basis for developing a sequent calculus for reasoning in W along the lines of Yetter's cyclic linear logic without exponentials [Yet90].
Computational complexity.As explained in Section 4, the decision procedure described in this paper provides an exponential upper bound for the computational complexity of deciding equations in the time warp algebra.A lower bound for this problem follows from the fact that the involution-free reduct of W generates the variety of distributive ℓ-monoids, which has a co-NP-complete equational theory [CGMS22].Tight complexity bounds have yet to be determined, however.8:19 Applications to graded modalities.The implementation described in Section 4 can be readily integrated in a type checker for a programming language with a graded modality whose gradings are first-order terms over W, generalizing previous work [Gua18].In broad strokes, such a type checker reduces each subtyping problem arising during type checking to a finite set of inequalities in the time warp algebra.The simplest interesting case is that of checking whether the type □ t A is a subtype of □ u A, which reduces to checking W |= u ≤ t.
The other cases in reducing type-checking problems to universal-algebraic problems depend on the exact language of types under consideration, including its subtyping relationship, which is beyond the scope of the current paper.Let us mention, however, that certain programming language features, such as the modeling of fixed computation rates, may require extending the language of gradings, and hence also the decision procedure.We discuss a simple example of such an extension in the next paragraph.
Extending the language.For certain applications, it may be profitable to extend the language of time warps with further operations.To illustrate, let us describe how 'definable' time warps can be added as constants to the language, while preserving decidability of the resulting equational theory.Let X be a countably infinite set of fresh variables for building first-order formulas.We call a time warp f definable if there exist first-order formulas ϕ f (x, y) and ψ f (z) in the signature τ = {⪯, S, 0, ω} with free variables x, y, and z, respectively such that for every A-valuation v : For example, the time warp ⊤ satisfying ⊤(0) = 0 and ⊤(n) = ω, for each n ∈ ω + \{0}, is definable by ϕ ⊤ (x, y) := (x ≈ 0 ⇒ y ≈ 0) ⋏ (0 ≺ x ⇒ y ≈ ω) and ψ ⊤ (z) := z ≈ S(0).Similarly, the time warp ⊥ satisfying ⊥(n) = 0, for each n ∈ ω + , is definable by ϕ ⊥ (x, y) := y ≈ 0 and ψ ⊥ (z) := z ≈ 0. Let D denote the set of definable time warps.Given any F = {f 1 , . . ., f n } ⊆ D, let W[F ] denote the algebra ⟨Res(ω + ), ∧, ∨, •, ⋆ , id, f 1 , . . ., f 1 ⟩.We call a term in the signature {•, ′ , e, f 1 , . . ., f n } F -basic and define F -samples analogously to samples by allowing F -basic terms in the construction.The notion of saturation is extended to F -sample sets, and following the proof of Lemma 3.2 shows that also the saturation of a finite F -sample set is finite.The notion of a diagram is then extended to saturated F -sample sets ∆ by defining the sets struct ∆ , log ∆ , inv ∆ as before, and also It is clear that Proposition 3.3 extends to saturated F -sample sets, using the definition of a definable time warp and the fact that homomorphisms from the term algebra into W[F ] map f to f , for each f ∈ F .Similarly, since every f ∈ F is completely described by the formulas ϕ f (x, y) and ψ f (z), also Proposition 3.9, extends to saturated F -sample sets.Hence, arguing as in Section 3, the equational theory of W[F ] is decidable.
A relational approach.We end this paper by discussing a different, relational point of view on time warps, which we believe will play a role in generalizing and extending the methods of this paper.It is well-known that, if L is a sufficiently well-behaved lattice, then any completely join-preserving function on L can be uniquely described via a certain binary relation on the set P of completely join-prime elements of L. The binary relations that occur in this way are exactly the distributors on P , when viewed as a posetal category.This raises the question as to whether the decision procedure we give here in the case L = ω + can be generalized to the algebraic structure of distributors on an object of a sufficiently well-behaved category P .In order to illustrate the idea, we explain here how the results of the paper can be alternatively understood in this relational language.
Let us write P = {1, 2, . . .} for the set of positive natural numbers, so that ω + is isomorphic to the lattice of downward closed subsets of P , via the function that sends x ∈ ω + to ↓x ∩ P .A time warp f : ω + → ω + may then be viewed alternatively as a binary relation R ⊆ P × P that is monotone (also called a weakening relation in, e.g., [GJ20]): that is, for any x, x ′ , y, y ′ ∈ P , if x ′ ≤ x, xRy, and y ≤ y ′ , then x ′ Ry ′ .Indeed, given a time warp f , the binary relation R f defined by where we note that the latter can also be written as the complementary relation of the converse of the relation R f .Hence, the lattice isomorphism between the lattice of time warps and M(P ) extends to an isomorphism between involutive residuated lattices, by equipping the lattice M(P ) with intersection, union, relational composition * , the unary operation (−) ⋆ given by R ⋆ := P 2 \(R −1 ), and the neutral element ≤ P .From the above considerations, since the isomorphisms are effective, it follows that questions about the theory of the structure W may be effectively translated into questions about the structure M(P ).Suppose that t is a term in the language of involutive residuated lattices that uses variables x 1 , . . ., x n .The above isomorphism allows us to translate the term t into a formula T (k, k ′ , X 1 , . . ., X n ) of second-order logic over the structure ⟨P, ≤⟩, where the X i are binary predicate symbols and k and k ′ are first-order variables.
≤ b :⇐⇒ a ∧ b = a, and, for all a, b, c ∈ L, a ≤ c/b ⇐⇒ ab ≤ c ⇐⇒ b ≤ a\c, where the residual operators are given explicitly for a, b ∈ L by a\b = {c ∈ L | ac ≤ b} and b/a = {c ∈ L | ca ≤ b}.
since time warps are determined by their values on ω\{0}.

Figure 2 :
Figure 2: Visualization of the extension of ⌊x⌋ δ

R
f := {(x, y) ∈ P 2 | x ≤ f (y)} is clearly monotone, and conversely, given a monotone binary relation R on P , the function f : P → ω + defined, for y ∈ P , by f (y) := {x ∈ P | xRy} is order-preserving and hence extends uniquely to a time warp by setting f (0) := 0 and f (ω) := p∈P f (p).These assignments f → R f and R → f R yield an order-isomorphism between the lattice of time warps and the lattice M(P ) of monotone binary relations on P , ordered by inclusion.Let us denote by * the associative operation of relational composition on M(P ), that is, R * S := {(x, z) ∈ P 2 | xRy and ySz for some y ∈ P }.Note that the natural order relation, ≤ P , on P is a neutral element in M(P ) for this composition operation * .Straightforward computations show that, for any f, g ∈ W , R f •g = R f * R g and R f ⋆ = {(x, y) ∈ P 2 | f (x) ≤ y − 1} = {(x, y) ∈ P 2 | y ≰ f (x)},