The causal sets program is an approach to quantum gravity. Its founding principles are that spacetime is fundamentally discrete (a collection of discrete spacetime points, called the elements of the causal set) and that spacetime events are related by a partial order. This partial order has the physical meaning of the causality relations between spacetime events.

## History

For some decades after the formulation of General Relativity, the attitude towards Lorentzian geometry was mostly dedicated to understanding its physical implications and not concerned with theoretical issues.[1] However, early attempts to use causality as a starting point have been provided by Weyl and Lorentz.[2] Alfred Robb in two books in 1914 and 1936 suggested an axiomatic framework where the causal precedence played a critical role.[1] The first explicit proposal of quantising the causal structure of spacetime is attributed by S. Surya[1] to Kronheimer and Penrose,[3] who invented Causal spaces in order to admit structures which can be very different from a manifold. Causal spaces are defined axiomatically, by considering not only causal precedence, but also chronological precedence.

The program of causal sets is based on a theorem[4] by David Malament, extending former results by E.C. Zeeman[5] and Hawking, King, McCarthy.[6][1] Malament Theorem states that if there is a bijective map between two past and future distinguishing space times that preserves their causal structure then the map is a conformal isomorphism. The conformal factor that is left undetermined is related to the volume of regions in the spacetime. This volume factor can be recovered by specifying a volume element for each space time point. The volume of a space time region could then be found by counting the number of points in that region.

Causal sets was initiated by Rafael Sorkin who continues to be the main proponent of the program. He has coined the slogan "Order + Number = Geometry" to characterize the above argument. The program provides a theory in which space time is fundamentally discrete while retaining local Lorentz invariance.

## Definition

A causal set (or causet) is a set ${\displaystyle C}$ with a partial order relation ${\displaystyle \preceq }$ that is

• Reflexive: For all ${\displaystyle x\in C}$, we have ${\displaystyle x\preceq x}$.
• Antisymmetric: For all ${\displaystyle x,y\in C}$, we have ${\displaystyle x\preceq y}$ and ${\displaystyle y\preceq x}$ implies ${\displaystyle x=y}$.
• Transitive: For all ${\displaystyle x,y,z\in C}$, we have ${\displaystyle x\preceq y}$ and ${\displaystyle y\preceq z}$ implies ${\displaystyle x\preceq z}$.
• Locally finite: For all ${\displaystyle x,z\in C}$, we have ${\displaystyle \{y\in C|x\preceq y\preceq z\))$ is a finite set.

We'll write ${\displaystyle x\prec y}$ if ${\displaystyle x\preceq y}$ and ${\displaystyle x\neq y}$.

The set ${\displaystyle C}$ represents the set of spacetime events and the order relation ${\displaystyle \preceq }$ represents the causal relationship between events (see causal structure for the analogous idea in a Lorentzian manifold).

Although this definition uses the reflexive convention we could have chosen the irreflexive convention in which the order relation is irreflexive and asymmetric.

The causal relation of a Lorentzian manifold (without closed causal curves) satisfies the first three conditions. It is the local finiteness condition that introduces spacetime discreteness.

## Comparison to the continuum

Given a causal set we may ask whether it can be embedded into a Lorentzian manifold. An embedding would be a map taking elements of the causal set into points in the manifold such that the order relation of the causal set matches the causal ordering of the manifold. A further criterion is needed however before the embedding is suitable. If, on average, the number of causal set elements mapped into a region of the manifold is proportional to the volume of the region then the embedding is said to be faithful. In this case we can consider the causal set to be 'manifold-like'.

A central conjecture of the causal set program, called the Hauptvermutung ('fundamental conjecture'), is that the same causal set cannot be faithfully embedded into two spacetimes that are not similar on large scales.

It is difficult to define this conjecture precisely because it is difficult to decide when two spacetimes are 'similar on large scales'. Modelling spacetime as a causal set would require us to restrict attention to those causal sets that are 'manifold-like'. Given a causal set this is a difficult property to determine.

### Sprinkling

The difficulty of determining whether a causal set can be embedded into a manifold can be approached from the other direction. We can create a causal set by sprinkling points into a Lorentzian manifold. By sprinkling points in proportion to the volume of the spacetime regions and using the causal order relations in the manifold to induce order relations between the sprinkled points, we can produce a causal set that (by construction) can be faithfully embedded into the manifold.

To maintain Lorentz invariance this sprinkling of points must be done randomly using a Poisson process. Thus the probability of sprinkling ${\displaystyle n}$ points into a region of volume ${\displaystyle V}$ is

${\displaystyle P(n)={\frac {(\rho V)^{n}e^{-\rho V)){n!))}$

where ${\displaystyle \rho }$ is the density of the sprinkling.

Sprinkling points as a regular lattice would not keep the number of points proportional to the region volume.

## Geometry

Some geometrical constructions in manifolds carry over to causal sets. When defining these we must remember to rely only on the causal set itself, not on any background spacetime into which it might be embedded. For an overview of these constructions, see.[7]

### Geodesics

A link in a causal set is a pair of elements ${\displaystyle x,y\in C}$ such that ${\displaystyle x\prec y}$ but with no ${\displaystyle z\in C}$ such that ${\displaystyle x\prec z\prec y}$.

A chain is a sequence of elements ${\displaystyle x_{0},x_{1},\ldots ,x_{n))$ such that ${\displaystyle x_{i}\prec x_{i+1))$ for ${\displaystyle i=0,\ldots ,n-1}$. The length of a chain is ${\displaystyle n}$. If every ${\displaystyle x_{i},x_{i+1))$ in the chain form a link, then the chain is called a path.

We can use this to define the notion of a geodesic between two causal set elements, provided they are order comparable, that is, causally connected (physically, this means they are time-like). A geodesic between two elements ${\displaystyle x\preceq y\in C}$ is a chain consisting only of links such that

1. ${\displaystyle x_{0}=x}$ and ${\displaystyle x_{n}=y}$
2. The length of the chain, ${\displaystyle n}$, is maximal over all chains from ${\displaystyle x}$ to ${\displaystyle y}$.

In general there can be more than one geodesic between two comparable elements.

Myrheim[8] first suggested that the length of such a geodesic should be directly proportional to the proper time along a timelike geodesic joining the two spacetime points. Tests of this conjecture have been made using causal sets generated from sprinklings into flat spacetimes. The proportionality has been shown to hold and is conjectured to hold for sprinklings in curved spacetimes too.

### Dimension estimators

Much work has been done in estimating the manifold dimension of a causal set. This involves algorithms using the causal set aiming to give the dimension of the manifold into which it can be faithfully embedded. The algorithms developed so far are based on finding the dimension of a Minkowski spacetime into which the causal set can be faithfully embedded.

• Myrheim–Meyer dimension

This approach relies on estimating the number of ${\displaystyle k}$-length chains present in a sprinkling into ${\displaystyle d}$-dimensional Minkowski spacetime. Counting the number of ${\displaystyle k}$-length chains in the causal set then allows an estimate for ${\displaystyle d}$ to be made.

• Midpoint-scaling dimension

This approach relies on the relationship between the proper time between two points in Minkowski spacetime and the volume of the spacetime interval between them. By computing the maximal chain length (to estimate the proper time) between two points ${\displaystyle x}$ and ${\displaystyle y}$ and counting the number of elements ${\displaystyle z}$ such that ${\displaystyle x\prec z\prec y}$ (to estimate the volume of the spacetime interval) the dimension of the spacetime can be calculated.

These estimators should give the correct dimension for causal sets generated by high-density sprinklings into ${\displaystyle d}$-dimensional Minkowski spacetime. Tests in conformally-flat spacetimes[9] have shown these two methods to be accurate.

## Dynamics

An ongoing task is to develop the correct dynamics for causal sets. These would provide a set of rules that determine which causal sets correspond to physically realistic spacetimes. The most popular approach to developing causal set dynamics is based on the sum-over-histories version of quantum mechanics. This approach would perform a "sum-over-causal sets" by growing a causal set one element at a time. Elements would be added according to quantum mechanical rules and interference would ensure a large manifold-like spacetime would dominate the contributions. The best model for dynamics at the moment is a classical model in which elements are added according to probabilities. This model, due to David Rideout and Rafael Sorkin, is known as classical sequential growth (CSG) dynamics.[10] The classical sequential growth model is a way to generate causal sets by adding new elements one after another. Rules for how new elements are added are specified and, depending on the parameters in the model, different causal sets result.

In analogy to the path integral formulation of quantum mechanics, one approach to developing a quantum dynamics for causal sets has been to apply an action principle in the sum-over-causal sets approach. Sorkin has proposed a discrete analogue for the d'Alembertian, which can in turn be used to define the Ricci curvature scalar and thereby the Benincasa–Dowker action on a causal set.[11][12] Monte-Carlo simulations have provided evidence for a continuum phase in 2D using the Benincasa–Dowker action.[13]

## References

1. ^ a b c d Surya, S. The causal set approach to quantum gravity. Living Rev Relativ 22, 5 (2019), p. 7-9. https://doi.org/10.1007/s41114-019-0023-1
2. ^ Bell JL, Korté H (2016) Hermann Weyl. In: Zalta EN (ed) The Stanford encyclopedia of philosophy, winter 2016 edn. Metaphysics Research Lab, Stanford University [1]
3. ^ Kronheimer E, Penrose R (1967) On the structure of causal spaces. Proc Camb Phil Soc 63:481
4. ^ Malament, David B. (July 1977). "The class of continuous timelike curves determines the topology of spacetime" (PDF). Journal of Mathematical Physics. 18 (7): 1399–1404. Bibcode:1977JMP....18.1399M. doi:10.1063/1.523436.
5. ^ E.C. Zeeman, Causality Implies the Lorentz Group, J. Math. Phys. 5: 490-493.
6. ^ Hawking S, King A, McCarthy P (1976) A new topology for curved space-time which incorporates the causal, differential, and conformal structures. J Math Phys 17:174–181.
7. ^ Brightwell, Graham; Gregory, Ruth (21 January 1991). "Structure of random discrete spacetime". Physical Review Letters. 66 (3): 260–263. Bibcode:1991PhRvL..66..260B. doi:10.1103/PhysRevLett.66.260. hdl:2060/19900019113. PMID 10043761. S2CID 32109929.
8. ^ J. Myrheim, CERN preprint TH-2538 (1978)
9. ^ Reid, David D. (30 January 2003). "Manifold dimension of a causal set: Tests in conformally flat spacetimes". Physical Review D. 67 (2): 024034. arXiv:gr-qc/0207103. Bibcode:2003PhRvD..67b4034R. doi:10.1103/PhysRevD.67.024034. S2CID 12748458.
10. ^ Rideout, D. P.; Sorkin, R. D. (2000). "Classical sequential growth dynamics for causal sets". Physical Review D. 61 (2): 024002. arXiv:gr-qc/9904062. Bibcode:1999PhRvD..61b4002R. doi:10.1103/PhysRevD.61.024002. S2CID 14652530.
11. ^ Sorkin, D. P. (20 March 2007). "Does Locality Fail at Intermediate Length-Scales". arXiv:gr-qc/0703099.
12. ^ Benincasa, D. M. T.; Dowker, F. (May 2010). "The Scalar Curvature of a Causal Set". Phys. Rev. Lett. 104 (18): 181301. arXiv:1001.2725. Bibcode:2010PhRvL.104r1301B. doi:10.1103/PhysRevLett.104.181301. PMID 20482164. S2CID 4560654.
13. ^ Surya, S. (July 2012). "Evidence for the continuum in 2D causal set quantum gravity". Classical and Quantum Gravity. 29 (13): 132001. arXiv:1110.6244. Bibcode:2012CQGra..29m2001S. doi:10.1088/0264-9381/29/13/132001. S2CID 118376808.

Introduction and reviews
Foundations
PhD theses
Talks
Manifoldness
• L. Bombelli, D.A. Meyer; The origin of Lorentzian geometry; Phys. Lett. A 141:226-228 (1989); (Manifoldness)
• L. Bombelli, R.D. Sorkin, When are Two Lorentzian Metrics close?, General Relativity and Gravitation, proceedings of the 12th International Conference on General Relativity and Gravitation, held July 2–8, 1989, in Boulder, Colorado, USA, under the auspices of the International Society on General Relativity and Gravitation, 1989, p. 220; (Closeness of Lorentzian manifolds)
• L. Bombelli, Causal sets and the closeness of Lorentzian manifolds, Relativity in General: proceedings of the Relativity Meeting "93, held September 7–10, 1993, in Salas, Asturias, Spain. Edited by J. Diaz Alonso, M. Lorente Paramo. ISBN 2-86332-168-4. Published by Editions Frontieres, 91192 Gif-sur-Yvette Cedex, France, 1994, p. 249; (Closeness of Lorentzian manifolds)
• L. Bombelli, Statistical Lorentzian geometry and the closeness of Lorentzian manifolds, J. Math. Phys.41:6944-6958 (2000); arXiv:gr-qc/0002053 (Closeness of Lorentzian manifolds, Manifoldness)
• A.R. Daughton, An investigation of the symmetric case of when causal sets can embed into manifolds, Class. Quantum Grav.15(11):3427-3434 (Nov., 1998) (Manifoldness)
• J. Henson, Constructing an interval of Minkowski space from a causal set, Class. Quantum Grav. 23 (2006) L29-L35; arXiv:gr-qc/0601069; (Continuum limit, Sprinkling)
• S. Major, D.P. Rideout, S. Surya, On Recovering Continuum Topology from a Causal Set, J.Math.Phys.48:032501, 2007; arXiv:gr-qc/0604124 (Continuum Topology)
• S. Major, D.P. Rideout, S. Surya; Spatial Hypersurfaces in Causal Set Cosmology; Class. Quantum Grav. 23 (2006) 4743-4752; arXiv:gr-qc/0506133v2; (Observables, Continuum topology)
• S. Major, D.P. Rideout, S. Surya, Stable Homology as an Indicator of Manifoldlikeness in Causal Set Theory, arXiv:0902.0434 (Continuum topology and homology)
• D.A. Meyer, The Dimension of Causal Sets I: Minkowski dimension, Syracuse University preprint (1988); (Dimension theory)
• D.A. Meyer, The Dimension of Causal Sets II: Hausdorff dimension, Syracuse University preprint (1988); (Dimension theory)
• D.A. Meyer, Spherical containment and the Minkowski dimension of partial orders, Order 10: 227-237 (1993); (Dimension theory)
• J. Noldus, A new topology on the space of Lorentzian metrics on a fixed manifold, Class. Quant. Grav 19: 6075-6107 (2002); (Closeness of Lorentzian manifolds)
• J. Noldus, A Lorentzian Gromov–Hausdorff notion of distance, Class. Quantum Grav. 21, 839-850, (2004); (Closeness of Lorentzian manifolds)
• D.D. Reid, Manifold dimension of a causal set: Tests in conformally flat spacetimes, Phys. Rev. D67 (2003) 024034; arXiv:gr-qc/0207103v2 (Dimension theory)
• S. Surya, Causal Set Topology; arXiv:0712.1648
Geometry
Cosmological constant prediction
• M. Ahmed, S. Dodelson, P.B. Greene, R.D. Sorkin, Everpresent lambda; Phys. Rev. D69, 103523, (2004) arXiv:astro-ph/0209274v1; (Cosmological Constant)
• Y. Jack Ng and H. van Dam, A small but nonzero cosmological constant; Int. J. Mod. Phys D. 10 : 49 (2001) arXiv:hep-th/9911102v3; (PreObservation Cosmological Constant)
• Y. Kuznetsov, On cosmological constant in Causal Set theory; arXiv:0706.0041
• R.D. Sorkin, A Modified Sum-Over-Histories for Gravity; reported in Highlights in gravitation and cosmology: Proceedings of the International Conference on Gravitation and Cosmology, Goa, India, 14–19 December 1987, edited by B. R. Iyer, Ajit Kembhavi, Jayant V. Narlikar, and C. V. Vishveshwara, see pages 184-186 in the article by D. Brill and L. Smolin: "Workshop on quantum gravity and new directions", pp 183–191 (Cambridge University Press, Cambridge, 1988); (PreObservation Cosmological Constant)
• R.D. Sorkin; On the Role of Time in the Sum-over-histories Framework for Gravity, paper presented to the conference on The History of Modern Gauge Theories, held Logan, Utah, July 1987; Int. J. Theor. Phys. 33 : 523-534 (1994); (PreObservation Cosmological Constant)
• R.D. Sorkin, First Steps with Causal Sets Archived 2013-09-30 at the Wayback Machine, in R. Cianci, R. de Ritis, M. Francaviglia, G. Marmo, C. Rubano, P. Scudellaro (eds.), General Relativity and Gravitational Physics (Proceedings of the Ninth Italian Conference of the same name, held Capri, Italy, September, 1990), pp. 68–90 (World Scientific, Singapore, 1991); (PreObservation Cosmological Constant)
• R.D. Sorkin; Forks in the Road, on the Way to Quantum Gravity, talk given at the conference entitled "Directions in General Relativity", held at College Park, Maryland, May, 1993; Int. J. Th. Phys. 36 : 2759–2781 (1997) arXiv:gr-qc/9706002; (PreObservation Cosmological Constant)
• R.D. Sorkin, Discrete Gravity; a series of lectures to the First Workshop on Mathematical Physics and Gravitation, held Oaxtepec, Mexico, Dec. 1995 (unpublished); (PreObservation Cosmological Constant)
• R.D. Sorkin, Big extra dimensions make Lambda too small; arXiv:gr-qc/0503057v1; (Cosmological Constant)
• R.D. Sorkin, Is the cosmological "constant" a nonlocal quantum residue of discreteness of the causal set type?; Proceedings of the PASCOS-07 Conference, July 2007, Imperial College London; arXiv:0710.1675; (Cosmological Constant)
• J. Zuntz, The CMB in a Causal Set Universe, arXiv:0711.2904 (CMB)
Lorentz and Poincaré invariance, phenomenology
• L. Bombelli, J. Henson, R.D. Sorkin; Discreteness without symmetry breaking: a theorem; arXiv:gr-qc/0605006v1; (Lorentz invariance, Sprinkling)
• F. Dowker, J. Henson, R.D. Sorkin, Quantum gravity phenomenology, Lorentz invariance and discreteness; Mod. Phys. Lett. A19, 1829–1840, (2004) arXiv:gr-qc/0311055v3; (Lorentz invariance, Phenomenology, Swerves)
• F. Dowker, J. Henson, R.D. Sorkin, Discreteness and the transmission of light from distant sources; arXiv:1009.3058 (Coherence of light, Phenomenology)
• J. Henson, Macroscopic observables and Lorentz violation in discrete quantum gravity; arXiv:gr-qc/0604040v1; (Lorentz invariance, Phenomenology)
• N. Kaloper, D. Mattingly, Low energy bounds on Poincaré violation in causal set theory; Phys. Rev. D 74, 106001 (2006) arXiv:astro-ph/0607485 (Poincaré invariance, Phenomenology)
• D. Mattingly, Causal sets and conservation laws in tests of Lorentz symmetry; Phys. Rev. D 77, 125021 (2008) arXiv:0709.0539 (Lorentz invariance, Phenomenology)
• L. Philpott, F. Dowker, R.D. Sorkin, Energy-momentum diffusion from spacetime discreteness; arXiv:0810.5591 (Phenomenology, Swerves)
Black hole entropy in causal set theory
• D. Dou, Black Hole Entropy as Causal Links; Fnd. of Phys, 33 2:279-296(18) (2003); arXiv:gr-qc/0302009v1 (Black hole entropy)
• D.P. Rideout, S. Zohren, Counting entropy in causal set quantum gravity; arXiv:gr-qc/0612074v1; (Black hole entropy)
• D.P. Rideout, S. Zohren, Evidence for an entropy bound from fundamentally discrete gravity; Class. Quantum Grav. 23 (2006) 6195-6213; arXiv:gr-qc/0606065v2 (Black hole entropy)
Locality and quantum field theory
Causal set dynamics
• M. Ahmed, D. Rideout, Indications of de Sitter Spacetime from Classical Sequential Growth Dynamics of Causal Sets; arXiv:0909.4771
• A.Ash, P. McDonald, Moment Problems and the Causal Set Approach to Quantum Gravity; J.Math.Phys. 44 (2003) 1666-1678; arXiv:gr-qc/0209020
• A.Ash, P. McDonald, Random partial orders, posts, and the causal set approach to discrete quantum gravity; J.Math.Phys. 46 (2005) 062502 (Analysis of number of posts in growth processes)
• D.M.T. Benincasa, F. Dowker, The Scalar Curvature of a Causal Set; arXiv:1001.2725; (Scalar curvature, actions)
• G. Brightwell; M. Luczak; Order-invariant Measures on Causal Sets; arXiv:0901.0240; (Measures on causal sets)
• G. Brightwell; M. Luczak; Order-invariant Measures on Fixed Causal Sets; arXiv:0901.0242; (Measures on causal sets)
• G. Brightwell, H.F. Dowker, R.S. Garcia, J. Henson, R.D. Sorkin; General covariance and the "problem of time" in a discrete cosmology; In ed. K. Bowden, Correlations:Proceedings of the ANPA 23 conference, August 16–21, 2001, Cambridge, England, pp. 1–17. Alternative Natural Philosophy Association, (2002).;arXiv:gr-qc/0202097; (Cosmology, Dynamics, Observables)
• G. Brightwell, H.F. Dowker, R.S. Garcia, J. Henson, R.D. Sorkin; "Observables" in causal set cosmology; Phys. Rev. D67, 084031, (2003); arXiv:gr-qc/0210061; (Cosmology, Dynamics, Observables)
• G. Brightwell, J. Henson, S. Surya; A 2D model of Causal Set Quantum Gravity: The emergence of the continuum; arXiv:0706.0375; (Quantum Dynamics, Toy Model)
• G.Brightwell, N. Georgiou; Continuum limits for classical sequential growth models University of Bristol preprint. (Dynamics)
• A. Criscuolo, H. Waelbroeck; Causal Set Dynamics: A Toy Model; Class. Quantum Grav.16:1817-1832 (1999); arXiv:gr-qc/9811088; (Quantum Dynamics, Toy Model)
• F. Dowker, S. Surya; Observables in extended percolation models of causal set cosmology;Class. Quantum Grav. 23, 1381-1390 (2006); arXiv:gr-qc/0504069v1; (Cosmology, Dynamics, Observables)
• M. Droste, Universal homogeneous causal sets, J. Math. Phys. 46, 122503 (2005); arXiv:gr-qc/0510118; (Past-finite causal sets)
• J. Henson, D. Rideout, R.D. Sorkin, S. Surya; Onset of the Asymptotic Regime for (Uniformly Random) Finite Orders; Experimental Mathematics 26, 3:253-266 (2017); (Cosmology, Dynamics)
• A.L. Krugly; Causal Set Dynamics and Elementary Particles; Int. J. Theo. Phys 41 1:1-37(2004);; (Quantum Dynamics)
• X. Martin, D. O'Connor, D.P. Rideout, R.D. Sorkin; On the "renormalization" transformations induced by cycles of expansion and contraction in causal set cosmology; Phys. Rev. D 63, 084026 (2001); arXiv:gr-qc/0009063 (Cosmology, Dynamics)
• D.A. Meyer; Spacetime Ising models; (UCSD preprint May 1993); (Quantum Dynamics)
• D.A. Meyer; Why do clocks tick?; General Relativity and Gravitation 25 9:893-900;; (Quantum Dynamics)
• I. Raptis; Quantum Space-Time as a Quantum Causal Set, arXiv:gr-qc/0201004v8
• D.P. Rideout, R.D. Sorkin; A classical sequential growth dynamics for causal sets, Phys. Rev. D, 6, 024002 (2000);arXiv:gr-qc/9904062 (Cosmology, Dynamics)
• D.P. Rideout, R.D. Sorkin; Evidence for a continuum limit in causal set dynamics Phys. Rev. D 63:104011, 2001; arXiv:gr-qc/0003117(Cosmology, Dynamics)
• R.D. Sorkin; Indications of causal set cosmology; Int. J. Theor. Ph. 39(7):1731-1736 (2000); arXiv:gr-qc/0003043; (Cosmology, Dynamics)
• R.D. Sorkin; Relativity theory does not imply that the future already exists: a counterexample; Relativity and the Dimensionality of the World, Vesselin Petkov (ed.) (Springer 2007, in press); arXiv:gr-qc/0703098v1; (Dynamics, Philosophy)
• M. Varadarajan, D.P. Rideout; A general solution for classical sequential growth dynamics of Causal Sets; Phys. Rev. D 73 (2006) 104021; arXiv:gr-qc/0504066v3; (Cosmology, Dynamics)
• M.R., Khoshbin-e-Khoshnazar (2013). "Binding Energy of the Very Early Universe: Abandoning Einstein for a Discretized Three–Torus Poset.A Proposal on the Origin of Dark Energy". Gravitation and Cosmology. 19 (2): 106–113. Bibcode:2013GrCo...19..106K. doi:10.1134/s0202289313020059. S2CID 121288092.;(Dynamics, Poset)