Algebra and Geometry with Computers R.Padmanabhan ...bremner/Padmanabhan_CMS2014... · It has been...

Gröbner Bases and Computer Algebra

Algebra and Geometry with Computers

R.PadmanabhanUniversity of Manitoba

1Wednesday, 11 June, 14

Abstract.Sir Roger Penrose once remarked that computers are intrinsically limited, compared to humans, when it comes to the doing of mathematics. Even those who think that such things can be proved may be interested in the empirical question: what kind of mathematics can computers do? Can a computer reason logically like humans and prove new theorems? A partial answer can be given using modern automated reasoning software such as Otter and Prover9. Otter, developed by William McCune at the Argonne National Laboratory, is the first widely used high-performance theorem prover. It is based on first-order inference rules, substitution principles, unification etc. Otter has since been replaced by Prover9, which is paired with Mace4, a counter-example generator. Both are available free for Mac OSX, Windows and the Unix platforms. In this presentation, we plan to give a brief survey of this relatively new area of experimental mathematics and give a list of ”new” theorems proved with the help of these theorem-provers. In particular, we plan to show some live demonstrations of automated deduction in the following areas of algebra and geometry: 1. Ring theory - commutativity theorems (in collaboration with Yang Zhang) 2. Cancellation Semigroup Conjecture (in collaboration with Iraghi Moghaddam) 3. Inverse semigroups (in collaboration with Michael Kinyon and J. Araujo) 4. Lattice theory - uniquely complemented lattices and Huntington laws 5. Projective planes as groups (using Maple routines) 6. Cayley-Bacharach implications (in collaboration with Bob Veroff).


Algebra and Geometry with Otter, Prover9 and MACE.

1. Ring theory - commutativity theorems (in collaboration with Yang Zhang)

2. Cancellation Semigroup Conjecture (in collaboration with Bill McCune, Bob Veroff and Iraghi Moghaddam)

3. Inverse semigroups (in collaboration with Michael Kinyon and J. Araujo)

4. Lattice theory - minimal axioms; uniquely complemented lattices and Huntington laws (in collaboration with Bill McCune and Robert Veroff)

5. Cayley-Bacharach implications (in collaboration with Bob Veroff).


A recent example of a theorem proved using Prover9 (2013)


Huntington Varieties of LatticesEdward V. HuntingtonSets of Independent Postulates for the Algebra of Logic Transactions of AMS, Vol. 5, No. 3 (1904) pp. 288-309



Is every uniquely complemented lattice distributive?



Is every uniquely complemented lattice distributive?

An affirmative answer to Huntington’s Problem would show that a lattice is Boolean if and only if it is uniquely complemented.


Huntington Varieties of LatticesIs every uniquely complemented lattice distributive?

In support of the Conjecture

Garrett Birkhoff and John von Neumann

G. Birkhoff and John von NeumannEvery uniquely complemented modular lattice is distributive (1939)


Huntington Varieties of LatticesIs every uniquely complemented lattice distributive?

In support of the Conjecture

Many other similar results followed in quick succession during the years 1938-44

Morgan Ward, J. E. McLaughlin and many others proved that atomic lattices, lattices of finite length, finite width, order reversability, de Morgan laws, algebraic lattices, semimodular lattices, ortholattices are all Huntington properties.


Huntington Varieties of LatticesIs every uniquely complemented lattice distributive?Enter 1945, a bombshell!


Huntington Varieties of LatticesIs every uniquely complemented lattice distributive?Enter 1945, a bombshell!

Dilworth, R. P. Lattices with unique complements. Trans. Amer. Math. Soc. 57, (1945). 123--154.It has been widely conjectured that every lattice with unique complements is a Boolean algebra. This conjecture is disproved; Using “free lattice” techniques, it is shown that every lattice is, in fact, a sublattice of a uniquely complemented lattice! -Garrett Birkhoff


http://www.ams.org.proxy2.lib.umanitoba.ca/msnmain?fn=130&fmt=hl&pg1=IID&s1=210821&v1=Dilworth%2C%20R%2E%20P%2E

http://www.ams.org.proxy2.lib.umanitoba.ca/msnmain?fn=130&fmt=hl&pg1=IID&s1=210821&v1=Dilworth%2C%20R%2E%20P%2E

http://www.ams.org.proxy2.lib.umanitoba.ca/msnmain?fn=305&pg1=CN&s1=Trans_Amer_Math_Soc&v1=Trans%2E%20Amer%2E%20Math%2E%20Soc%2E

http://www.ams.org.proxy2.lib.umanitoba.ca/msnmain?fn=305&pg1=CN&s1=Trans_Amer_Math_Soc&v1=Trans%2E%20Amer%2E%20Math%2E%20Soc%2E

Definition. K is a Dilworth variety if every lattice in K is imbeddable in a uniquely complemented lattice in K.Dilworth (1945): L is a Dilworth variety

Definition. K is a Huntington variety if every uniquely complemented lattice in K is distributive.Birkhoff, von Neumann (1940): The class of all modular lattices is a Huntington variety.




Adams, Sichler (1981): There exists a continuum of Dilworth varieties of lattices.




Adams, Sichler (1981): There exists a continuum of Dilworth varieties of lattices.

(2006) Here we prove that there exists a continuum of Huntington varieties of non-modular lattices.


An illustration of the implication of basic gL22Wednesday, 11 June, 14

An application of Skolem function in Prover9 (with Bob Veroff)

formula for thegroup inverse

captured by Prover9


Thank You


Algebra and Geometry with Computers R.Padmanabhan ...bremner/Padmanabhan_CMS2014... · It has been...

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