1. Construction techniques in topological universal algebra
Wolfram Bentz
St. Francis Xavier University
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2. Introduction
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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)
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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
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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)
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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
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7. Separation Conditions:b
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8. Separation Conditions:b
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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}
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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 p’s are ternary term functions: 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
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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 p’s would be Hagemann-Mitschke terms.
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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
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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
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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
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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 form a = p1(x1), p1(y1) = p2(x2), p2(y2) = p3(x3), …, pn-1(yn-1) = pn-1(xn-1), pn(yn) = b where the p’s are unary polynomial functions and the x’s and y’s are in D
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16. Swierczkowski – Extension
For any two elements a and b of F, look at connections of the form a = p1(x1), p1(y1) = p2(x2), p2(y2) = p3(x3), …, pn-1(yn-1) = pn-1(xn-1), pn(yn) = b where the p’s are unary polynomial functions and the x’s and y’s are in D
To each such connection assign the value Σd(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
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17. Swierczkowski – Extension
The Swierczkowski extension allows one to construct a topological algebra having a prescribed subspace, provided the subspace is metrizable
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18. Example
When examining homotopy properties, Taylor used the free algebra over this space
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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
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20. Separation Conditions:b c D
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21. Example
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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
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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
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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
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25. Example (non-Hausdorffness)
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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
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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
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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
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29. Thank you!
Questions?
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