Construction techniques in topological universal algebra

1
Construction techniques in topological universal
algebra

• Wolfram Bentz
• St. Francis Xavier University

2
Introduction
3
Universal Topological Algebra

• Algebra and Topology are compatible if all
  functions are continuous
• Via compatibility the algebraic structure
  restricts the topological one and vice versa
• General Question How do algebraic properties
  correspond to topological ones?
• Typical Results have the form The varieties with
  (A) are exactly those whose topological algebras
  satisfy (T)

4
The construction task

• To draw topological conclusions from algebraic
  properties, one applies continuity directly to
  algebraic conditions
• No such direct way is apparent when trying to
  deduce algebraic conclusions from topological
  properties
• Hence, such results are proved by finding
  counterexamples

5
The construction task

• Counterexamples for single varieties can show
  that a correspondence does not exist
• To get a more general result one needs to find a
  general construction principle, working for all
  varieties satisfying an algebraic property (a
  generic counterexample)

6
Example

• Coleman (1997) A variety satisfies if and only
  if it is n-permutable for some n
• One direction already shown in Gumm (1984)
  n-permutable varieties satisfy

7
Separation Conditions
A
8
Separation Conditions
A
9
Generic Counterexample

• Note the original proof does not construct a
  counterexample directly the following has been
  derived from it and various preceding results
• Let V be a variety that is not n-permutable for
  any n.
• Consider F(x,y), the free algebra in V over x,y

10
Generic Counterexample

• For any element a in F(x,y) consider the elements
  b reachable from a by the following chain of
  equations, where the ps are ternary term
  functionsa p1(x,y,y), p1(x,x,y) p2(x,y,y),
  p2(x,x,y) p3(x,y,y), , pn-1(x,x,y)
  pn-1(x,y,y), pn(x,x,y) b

11
Generic Counterexample

• a p1(x,y,y), p1(x,x,y) p2(x,y,y), p2(x,x,y)
  p3(x,y,y), , pn-1(x,x,y) pn-1(x,y,y),
  pn(x,x,y) b
• Note that b x is reachable from a y (choose n
  1, p1(u,v,w) v), but if b y would be
  reachable from a x, then the ps would be
  Hagemann-Mitschke terms.

12
Generic Counterexample

• Now let a subset U of F(x,y) be open if for any
  element a in U, it also contains the elements
  reachable from a
• These sets form a compatible topology
• Every open set containing y must contain x.
  Conversely, the set of all elements reachable
  from x is open, but does not contain y

13
Generic Counterexample

• So x and y are not separable in the T1 sense.
• Identifying all elements that are contained in
  exactly the same open sets is a congruence
• The resulting quotient is T0, but preserves the
  T1-inseparability between x and y

14
More general examples

• The preceding construction relies heavily on the
  Hagemann Mitschke terms for n-permutability
• This was possible because an exact algebraic
  characterization is known
• If this is not the case, look for a topological
  construction, that might not be compatible in
  general
• If it is close to a compatible construction,
  one can examine the cases were compatibility is
  achieved

15
Swierczkowski Extension

• A metric construction by Taylor (based on a
  topological one by Swieczkowski)
• Consider a space D with metric d, and the free
  algebra F over D (in a variety)
• For any two elements a and b of F, look at
  connections of the forma p1(x1), p1(y1)
  p2(x2), p2(y2) p3(x3), , pn-1(yn-1)
  pn-1(xn-1), pn(yn) bwhere the ps are unary
  polynomial functions and the xs and ys are in D

16
Swierczkowski Extension

• For any two elements a and b of F, look at
  connections of the forma p1(x1), p1(y1)
  p2(x2), p2(y2) p3(x3), , pn-1(yn-1)
  pn-1(xn-1), pn(yn) bwhere the ps are unary
  polynomial functions and the xs and ys are in D
• To each such connection assign the value
  Sd(xi,yi)
• Set d(a,b) to be the inf of all corresponding
  connection values, then d is a compatible metric
  on F extending the metric of D

17
Swierczkowski Extension

• The Swierczkowski extension allows one to
  construct a topological algebra having a
  prescribed subspace, provided the subspace is
  metrizable

18
Example

• When examining homotopy properties, Taylor used
  the free algebra over this space

19
Swierczkowski Extension

• If the desired counterexamples are not metrizable
  one can modify a Swierczkowski Extension
• Coleman (1996) congruence permutable varieties
  do not necessarily satisfy

20
Separation Conditions
A
21
Example


22
Example (Coleman)

• Take the free algebra F over the real numbers
• Extend the metric topology of the reals to all of
  F (Swierczkowski)
• Enlarge the topology so that the subalgebra
  generated by Q is closed
• This topology satisfies T1, but fails T3
• Examine whether continuity still holds in the new
  topology

23
Example

• This construction works (as a topological space)
  for every non-trivial variety
• It appears likely that this is fundamental
  construction in the sense that it is a
  topological algebra in any variety failing

24
Example

• Using this example, a large class of varieties
  was characterized
• Bentz (2007) A depth 1 variety V satisfies
  if and only if V is trivial
• Note depth 1 is a restriction on the defining
  equations of a variety

25
Example (non-Hausdorffness)

26
Example of a partial construction (Coleman)

• Take the free algebra F over the real numbers
• Extend the metric topology of the reals to all of
  F (Swierczkowski)
• Introduce an extra point that is not
  Hausdorff-separable from some base point in F.
• Try to fit the algebraic structure

27
Extending the algebra

• Involving the extra point in the operations must
  preserve the laws of the variety
• Everything must stay continuous with respect to
  the new topology

28
Results with this construction

• Coleman congruence 4-permutability is not strong
  enough to force
• A Depth 1 variety satisfies if and only if it is
  congruence modular and n-permutable for some n
  (Bentz, 2006)
• This partial construction unfortunately will not
  work in all cases
• More doubled points might be a promising approach

29
Thank you!
Questions?