PROPERTIES OF COMPLEMENTS IN THE LATTICE OF CONVERGENCE STRUCTURES

Relative complements and differences are investigated for several convergence structure lattices, especially the lattices of Kent convergence structures and the lattice of pretopologies. Convergence space properties preserved by relative complementation are studied. Mappings of some convergence structure lattices into related lattices of lattice homomorphisms are considered.

and quotients.
The definitions used are essentially those of [3], [4] and [ii] with a convergence structure on X considered as a map q X + 8(F(X)) (the power set of the set of filters on X). U(X) and are the set of ultrafilters on X and the prlncipal ultrafilter generated by {x}. For a convergence space (X,q), let %q, q and q be the topological, pretopological, and complety regula topological modifi-COS of q. The q-//mlt Set of a filter F is ad q(F) (x F E q(x)} and the cosue cl(A) of a subset A is {x F q(x) for some filter F with A F}.
An element z of lattice L is the pseudo-complement of x rzive to y (x,y) if z is the greatest element with x^z<y and the pseudo-difference of y and x (y-x) if z is the least element with y<xvz. If L is a complete lattice with 0 and i the least and greatest elements, then the pseudo-complement of x is x* x,0 and the pseudo-difference of x is -x l-x (a change of notation from [8]).
The relative pseudo-complement and pseudo-difference of two convergence structures in C(X) were described in [8] as: q,r(x) {FI F or G n _c F for some G r(x) \q(x)} q-r(x) {F F + G B(X) or is in q(x) for all G r(x)} For q and r limitierungs of Fischer [3] or pseudo-topologies the same descriptions hold for relative pseudo-complements and pseudo-differences in the lattices of limitierungs or pseudotopologies. In P(X), the lattices of pretopologles on X, pseudo-differences do not exist from [9]. Relative pseudo-complements also fall to exist in P(X) even though, from [9], P(X) is pseudo-complemented. EXAMPLE 2.1: On an infinite set X, let A be an infinite subset with infinite complement and x X. If q(r) is the finest pretopology on X such that an ultrafilter F q(r)-converges to x if and only if F or F is free and contains A(X-A then q,r does not exist in P(X).
Many properties of convergence structures are preserved by relative pseudocomplementatlon. If r is a limitierung (resp. pseudotopology, pretopology, topology), then from [8], q,r is the same type of structure for any convergence structure q. PROPOSITION 2.2: For any convergence structures q and r on X: (i) q,r is T I (Hausdorff) if r is T 1 (Hausdorff).
(ii) q,r is T 3 if r is T 3.
(ill) q,r is compact if and only if r is compact and for any ultrafilter F, ad (G) ad (G) for some G c F. r q (iv) q,r is T-regular [5] if r is T-regular.
(vi) q,r is second countable if r is second countable and q,r has at most countably many discrete points.
In addition, for any pretopology q on X: (vii) q,r is a completely regular topology if r is a completely regular topology.
(viii) q,r is m-regular [6] if r is m-regular.
(ix) q,r is C-embedded [6] if r is C-embedded.
(ii) If F E q,r(x) with G c F for some G r(x) \ q(x) then cl G E r(x) \ q(x) r since r is regular so cl G ccl G ccl F and cl F q,r-converges r q,r q,r q,r to x. If q,r(x) {} then q,r is T I so Clq,r .
(v) If F E q,r(x) with G E F and G E r(x) \q(x), then H r(x) \q(x) for some H with filterbase of cardlnallty less than for any cardinal Let B be a countable basis for (X,r). Then B' u {x q,r(x) {}} is a countable basis for (X,q,r).
(vii) Since q,r is topological from [8], suppose A is q,r-closed and x A.
Then if x cl (A), any real valued continuous function on (X,r) which r separates x and A is also q,r-continuous. If x cl (A), then q,r is r discrete at x so x and A can be separated by a q,r-continuous, realvalued function.
(viii) If F q,r(x) and G r(x) \ q(x) with cl G E r(x) \ q(x) then from 0r (vii), if r is the completely regular modification of r, cl G = r cl G c cl (q,r)G = cl(q,r F and if q,r is discrete at x, so is q,r (q,r) and the conclusion follows.
(ix) From [8], q,r is pseudo-topological if r is pseudo-topological. If r is Hausdorff and u-regular, then q,r has the same properties from (i) and (viii) so by [6], q,r is C-embedded if r is C-embedded. COROLLARY 2.3: (i) If r is the finest first countable structure coarser than r, then (q,r) q,r for every convergence structure q.
(ii) If Rr, the finest regular structure coarser than r [7], is T I, then q,Rr < R(q,r).
o o q*r o PROOF: Since r < r, q,r < q,r and being first countable implies q,r As q,Rr is T 3 from (ii) of Proposition 2.2 and q,Rr < q,r, then q,Rr <_ R(q,r). x. Then 0 is regular but not T I and q* is not regular.
(ii) Let r be a convergence structure on an infinite set X for which Rr # r and Rr is TI, such as a non-regular T2-convergence structure which is finer than some T2, regular topology. Then 1 R(r,r) # r,Rr.
The following description of the convergent ultrafilters of the pseudo-dlfference q-r of two convergence structures is given in [8]: LEMMA 2.5: An ultrafilter F q-r converges to x if and only if F q-converges to x or does not r-converge to x.
Because q-r can have so many convergent ultrafilters, most convergence space properties are not preserved. This can also be observed from the result of [9] that the image of the map q + l-q is the lattice of pseudotopologles. For example, one can readily show that q-r is not pretopologlcal if q r, r is T 1 and q is not discrete and l-q is not regular if q is T I and not discrete. A few properties can be easily seen to be preserved. PROPOSITION 2.6: For any convergence structures q and r on X, (1) q-r is T 1 if and only if q is T 1 and the pretopologlcal modification r of r is indiscrete.
(il) q-r is Hausdorff if and only if l-q < r.
(ill) q-r is compact if q is compact.
(iv) q-r is compact if and only if no ultrafilter F (l-r)^q-converges to every point. (1) (Hqe), (Hre) < H (q,re)-(li) F converges to x (xe) with respect to (Hqe),(Hre) if and only if py(F) q,ry-converges to x for some y e F.
(iii) (Hqe),(re) Hw(qe,re). (v) (q)* * Hwq e PROOF: (i) If F H(qe,re)-converges to x (x e) then each Pc(F) qa,ra-converges to x so for each there exists a filter G on X with G Pc(F) xe or G _c p(F) and Ge re(x=) \ qe(xe)" Then HGe is Hr,-convergent to x and HGe (Hra)(x) \ (Hqe)(x) or HG x and converges to x since HGe -c Hp(F) _c F.
for some y so py (iii) follows immediately from (ii) and the definition of a weak product.
(iv) If IF[ > 1 and Hqe < Hwqe then for F any filter on X and x X let (v) is a direct consequence of (iv) since q* q*O. The product operation can also be viewed as a lattice operation on C(X). (NXa,Hq) is a complete join homomorphism.
If (X,q) is a convergence space with an equivalence relation on X, let X/b e the quotient space with quotient structure q, A {x x A} for A X and, for F a filter on X, { A e F}. Let f" C(X) C(X/) be the map f(q) q. (ii) For any q and r in C(X), fA(q,r) fA(q),fA(r) and fA(q-r)= fA(q)-fA(r).
As one would expect, Proposition 2.12 establishes that the restriction of the relative complements to a subspace are the complements of the restrictions.

LATTICE OPERATORS INDUCED BY RELATIVE COMPLEMENTS.
The relative pseudo-complement and pseudo-difference induce four obvious self-maps of C(X) for each convergence structure q: (i) f*(q)" f*(q)(r) q,r (ii) f,(q)" f,(q)(r) r,q (iii) f (q)" f (q)(r)= q-r (iv) f (q)" f (q)(r)= r-q Of these maps, (i) and (iv) were considered in [8]. Only (i) and (iv) will be considered here since (ii) and (iii) have similar lattice properties if considered as maps of C(X) into its dual.
If F is a cardinal, a subset A of a lattice L is prime with rpect to Fjoins in L if for any subset {x Y F} with vx A, some x e A. A conver-Y gence structure q of C(X) is join prime if each q(x)\ {x} is prime with respect to finite joins in {r(x) \ {} r C(X)}. As an extension of a result in [8] one has PROPOSITION 3.1: For any convergence structure q on X: (i) f"(q) is a complete meet homomorphism.
(ii) f*(q) is a F-join homomorphism if and only if q(x) \ {x} is prime with respect to F-joins in F(X) for any cardinal F.
(iii) f*(q) is bijective if and only if q is discrete.
PROOF: (i) is a result of [8] while the proof of (ii) parallels the result of [8] for finite joins. (iii) is a property of complete lattices. PROPOSITION 3.2: (i) f_(q) is a complete join homomorphism.
(ii) f_(q) is complete with respect to F-meets for a cardinal F if and only if each q(x) is complete with respect to F-meets in C(X) for each x.
(iii) f (q) is bijective if and only if q is indiscrete.
PROOF: (i) is from [8] while the proof of (ii) is similar to Theorem 4.2 of [8]. (iii) is dual to Proposition 3.1(iii).
From Proposition 3.2 one can observe that f (q) is a complete lattice homomorphism if and only if q is a pretopology. If F and are infinite cardinals with F < , by choosing the cardinal of X large enough so that if y e X and q(x) is discrete for y # x and q(y) is closed with respect to F-meets but not -meets, then f (q) is a F-homomorphism that is not an -homomorphism.
Using the given four lattice operators, one can construct maps of certain sublattices of C(X) into the duals of their lattices of homomorphisms (with coordinatewise order). For example, if L(X) is the lattice of limitierungs on X and P(X) the lattice of pretopologies, one can define fL:* L(X) L L by fL*(q)(r) q,r and f P(X) + similarly, where LL(pP) is the dual of the lattice of homomorphisms of L(X) and q,r is the relative pseudo-complement in C(X). The succeeding two propositions follow directly from the definitions and properties of pseudo-complements and differences. (ii) f (q) is continuous. PROPOSITION 3.6: fL and fp are continuous in the induced topologies.
Since join-prime elements q determine when f*(q) is a homomorphism, one may note that if q(x) is join-prime, there exists at most one ultrafilter F not qconvergent to x. Also, the join-prime elements of C(X) form ameet-sublattice of C(X) but not a join-sublattice.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall. This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under different approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos). We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/ according to the following timetable:

Manuscript Due
December 1, 2008 First Round of Reviews March 1, 2009