Closure Spaces: A Brief Overview
The concept of uniquely generated closure spaces has begun
to be studied as a common thread emerging in computer applications, in
graphs, and in discrete geometries. Briefly, a closure operator CL is said
to be
uniquely generated if for any closed set there a unique minimal
subset for which this is the closure. Such a closure operator acting on
a set, or universe, of elements, U is said to be a
closure space
(U, CL).
The importance of uniquely generated closure spaces lies
in the fact that in discrete systems they play a role that is in many respects
analogous to the vector spaces of classical mathematics.
The basis set of a vector space can be generalized as
a
matroid and the vector space is a closure (or spanning) operator
on this basis set which satisfies the SteinitzMacLane exchange axioms.
Every uniquely generated closure operator can be shown to satisfy the antiexchange
axioms, and therefore be the closure of an
antimatroid. Consequently
a closure space is to an antimatroid, what a vector space is to a matroid.
But there the analogy stops. The properties of closure
spaces are very different.
Closure operators are fairly common, although they frequently
have other names, for example ``convexity''. The convex hull of a discrete
set is an uniquely generated closure. A theory of convex geometries is
developed in [EJ85]. Convexity in graphs has been examined in [P71], [FJ86].
The ``lower ideals'', or ``down sets'' of a partially ordered set are closed.
In concurrent computing, the concept of a ``transaction'' is a simple closure
operator. Algorithmic closure, in particular that of greedy algorithms
is found in [KLS91] which introduces the term ``greedoid'', a special kind
of antimatroid.
In the case of networks, or undirected graphs, a definition of ``neighborhood
closure'' has proven to be quite useful.
It is neither ``matroid'' nor ``antimatroid''.
(See our major heading on ``Social, and other, Networks'' for more details''.
Most closure operators are unary; they have a single argument, usually
a set.
``Galois closure'', which usually defined closure relations between a
set of objects and their properties, is binary.
(See our major heading on ``Concept Graphs and Logical Inference''.)
The subsets of a closure space can be partially ordered
to create a lattice with many interesting properties [P96]. An unusual
property comes from the observation that for any set Z in U, { X  X.CL
= Z.CL } must be a power of 2. Thus any uniquely generated closure operator
CL partitions the subsets of U into a disjoint collection of subsets, each
containing a single closed set and each consisting of 2^k subsets. Hence
every closure space on n points has a characteristic signature of the form
< a_n, ... a_1, a_0 > where a_k counts the number of closure subsets
of size 2^k. The a_k are called a partition of 2^n. Since every
such partition gives rise to at least one closure space, the number of
such partitions gives a lower bound for the number of closure spaces on
n elements [P95a].
More recent work has included the definition of distance
in antimatroids [P97], which turns out to be nonlocal; transformations
of antimatroid closure spaces [P98a], which includes a definition of continuity
in discrete spaces; and uniform closure spaces [P98b].
Related Papers

[EJ85] P.H. Edelman & R.E. Jamison, The theory of convex
geometries,
Geometriae Dedicata, 19(3):247270, Dec. 1985.

[FJ86] M. Farber & R.E. Jamison, Convexity in graphs
and hypergraphs,
SIAM J. Algebra and Discrete Methods, 7(3):433444,
July 1986.

[KLS91] B. Korte, L. Lovasz & R. Schrader,
Greedoids,
SpringerVerlag, Berlin, 1991.

[P71] J.L. Pfaltz, Convexity in directed graphs,
J. of
Comb. Theory, 10(2):143162, Apr. 1971.

[P95a] J.L. Pfaltz,
Partitions of
2^n,
Congressus Numerantium 109:312, 1995.

[P95b] J.L. Pfaltz,
Partition Coefficients
of Acyclic Graphs,
21st Intern'l Workshop on Graph Theoretic Concepts
in Computer Science, Aachen, June 1995 (Springer Verlag, LNCS #1017)
313332.

[P96] J.L. Pfaltz,
Closure Lattices,
Discrete Mathematics ,
154:217236, June 1996.

[PKM97] J.L. Pfaltz, John Karro, Sean McCulloch,
Distance
in AntiMatroids,
Congressus Numerantium 127:522, 1997.

[PK98a] J.L. Pfaltz, John Karro,
Transformations
of Antimatroid Closure Spaces,
Computer Science TR 98013 June
1998.

[PJ00] J.L. Pfaltz, Robert E. Jamison
Closure
Systems and their Structure,
5th Intern'l Seminar on Relational
Methods in Computer Science, RelMiCS 2000, Valcartier, Quebec, Jan.
2000, 121123.
Also found in
Information Sciences, 139(2001) 275286

[JP05] Robert E. Jamison, J.L. Pfaltz,
Closure
Spaces that are not Uniquely Generated,
Discrete Applied Math,
147, (2005), 6999,
also in,
Ordinal and Symbolic Data Analysis, OSDA 2000
Brussels, Belgium July 2000.

[P01] J.L. Pfaltz,
Transformations of Concept
Graphs: An Approach to Empirical Induction ,
2nd Intern'l Workshop on Graph Transformation and Visual Modeling Techniques, GTVM 2001,
Satellite Workshop of ICALP 2001, Crete, Greece July 2001, 320326.

[PT02a] J.L. Pfaltz, C. M. Taylor
Closed Set Mining of Biological Data,
BIOKDD 2002, Workshop on Data Mining in Bioinformatics,
(at KDD 2002, Edmonton, Alberta,
July 2002, 4348.

[PT02b] J.L. Pfaltz, C. M. Taylor
Scientific Discovery through Iterative Transformations
of Concept Lattices,
Workshop on Discrete Mathematics and Data Mining,
(at 2nd SIAM Conf. on Data Mining, Arlington VA)
April 2002, 6574.

[P04a] J.L. Pfaltz,
A Category of Discrete Closure Systems,
Spatial representation:Discrete vs. Continuous Computational Models ,
Dagstuhl Seminar 04351, August 2004.

[P04b] J.L. Pfaltz,
A Category of Discrete Partially Ordered Sets,
MidAtlantic Algebra Conf.
George Mason Univ., Fairfax VA, Nov. 2004.

[KP04] Ralph Kopperman and J.L. Pfaltz,
Jordan Surfaces in Discrete Topologies,
IWCIS, 10th Intern'l Workshop on Combinatorial Image Analysis,
Auckland, NZ, Nov. 2004.

[P06a] J.L. Pfaltz,
Using Concept Lattices to Uncover Causal Dependencies in Software,
Proc. Int. Conf. on Formal Concept Analysis, Springer LNAI 3874 ,
Dresden, 233247, Feb.. 2006.

[P06b] J.L. Pfaltz,
Logical Implication and Causal Dependency,
ICCS, 14th International Conference on Conceptual Structures,
Springer LNAI 4068,
Aalborg University, Denmark, July 2006, pp 145157 (supplemental volume).

[P07] J.L. Pfaltz,
What Constitutes a Scientific Database?,
SSDBM, 19th International Conference on Scientific and Statistical
Database Management,,
Banff, Canada, July 2007.

[P07] J.L. Pfaltz,
Establishing Logical Rules from Empirical Data
ICTAI, 19th International Conference on Tools with Artificial Intelligence
,
Patras, Greece, Oct. 2007.

[P08] J.L. Pfaltz,
Establishing Logical Rules from Empirical Data
Intern. Journal on Artificial Intelligence Tools
,
Vol. 17, no. 5 (2008) 9851001

[P11] J.L. Pfaltz,
Mathematical Continuity in Dynamic Social Networks
Third Intern. Conf. on Social Informatics, SocInfo 2011
,
LNCS #6984, (3650),
Singapore, 2011.
Reprinted in "Social Network Analysis and Mining", 2013.

[PS12] J.L. Pfaltz and Josef Slapal,
Transformations of discrete closure systems
Acta Math. Hungar.
,
Sept.2012 (electronic version), March 2013 (print version).

[P12] J.L. Pfaltz,
Finding the Mule in the Network
Intern. Conf. on Advances in Social Network Analysis and Mining, ASONAM 2012
,
Istanbul, Turkey, August 2012,
9851001

[P13a] J.L. Pfaltz,
The Irreducible Spine(s) of Discrete Networks
Web Information Systems Engineering  WISW 2013
,
LNCS #6984, Part 2, (104117),
Nanjing, China, Oct. 2013,

[P13b] J.L. Pfaltz,
Mathematical Evolution in Discrete Networks
Mathematics for Applications
,
Vol. 2, (2013) 153167

[P13b] J.L. Pfaltz,
Continuous Functions Over Discrete Partial Orders
Mathematics for Applications
,
Vol. 4, (2015) 6176