S. Winker. Absorption and idempotency criteria for a problem in near-Boolean algebras. Journal of Algebra, 153:414--423, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....0] by 24) z) by 17) The proof of Theorem 27 was discovered by Don Monk. B # 4 . 6 Proof. z) by B 4 ) z) B. Proof. By the definition of 1 and Theorems 18, 19, 25, 26, 21, 27, and 28. 5 R W 1 In 1992, Winker [5] proved that each of the following axioms is a su#cient condition for a Robbins algebra to satisfy the Huntington equation, and therefore to be a Boolean algebra. W 2 ) x = x (double negation) W 1 ) x 0 = x (zero) a = a (idempotent) b = b (absorption) b = b (absorption within negation) ....

S. Winker. Absorption and idempotency criteria for a problem in near-Boolean algebras. Journal of Algebra, 153:414--423, 1992.

....his students [Henkin et al. 1971; Tarski, private communication 1980] to no avail. Subsequently, S. Winker attacked the problem using a combination of individual insight and an automated reasoning program. He was able to prove that certain conditions suffice to make a Robbins algebra Boolean [Winker, 1990, 1992]. But, with the reasoning programs available, he was unable to prove that any of these conditions follow from the axioms. Then, in 1996, with the development of a new theorem prover called EQP [McCune, 1997b] the problem was cracked. The 133 step solution, which relied on a technique known as ....

S. Winker, Absorption and idempotency criteria for a problem in near-Boolean algebras. J. Algebra, 153(2):414--23, 1992.

....subterms of the terms in Luka 5. A representative member of Luka 5 is i(n(i(x; y) i(n(i(i(z; y) n(z) u) Robbins. The Robbins set is derived from a theorem on the relationship between Robbins and Boolean algebras: that the property 9x9y (x y = y) makes a Robbins algebra Boolean [21]. The set Robbins is the left sides of the first 2000 rewrite rules derived in a KnuthBendix search with OTTER. The operator is associative and commutative, which we handle with axioms and rewrite rules rather than with AC unification. No computer has yet proved the theorem. The indexed set is ....

S. Winker. Absorption and idempotency criteria for a problem in near-boolean algebras. Preprint MCS-P177-0990, Argonne National Laboratory, Argonne, IL, August 1990.

.... Robbins problem [101] In the early Nineties, Steve Winker proved by hand that each of the following two conditions #x#y x y = x (FWC) #x#y n(x y) n(x) SWC) termed First Winker Condition and Second Winker Condition, respectively, in [80] is su#cient to make a Robbins algebra Boolean [97, 98], but such lemmas remained beyond the possibilities of automated theorem provers. In 1996 the automated prover EQP of William McCune proved that Robbins algebras are Boolean, as reported in [81] The proof was obtained by showing that: The First Winker Condition implies the Huntington axiom ....

Steve Winker. Absorption and idempotency criteria for a problem in near-Boolean algebras. Journal of Algebra, 153(2):414--423, 1992.

....8x; x x = x, 2) 9c8x; c x = x, and (3) 9c8x; c x = c. Winker then proved (by hand, with insight from theorem prover searches) several weaker conditions sufficient. The two such conditions that play a role in the present work are contained in the following two lemmas. Lemma 1 (S. Winker [13, 14]) A Robbins algebra satisfying 9c9d; c d = c is a Boolean algebra. Lemma 2 (S. Winker [13, 14] A Robbins algebra satisfying 9c9d; n(c d) n(c) is a Boolean algebra. Appendix B contains a computer proof of Lemma 1. Note that Lemma 2 is a strengthening of Lemma 1. 2. The Solution This ....

....insight from theorem prover searches) several weaker conditions sufficient. The two such conditions that play a role in the present work are contained in the following two lemmas. Lemma 1 (S. Winker [13, 14] A Robbins algebra satisfying 9c9d; c d = c is a Boolean algebra. Lemma 2 (S. Winker [13, 14]) A Robbins algebra satisfying 9c9d; n(c d) n(c) is a Boolean algebra. Appendix B contains a computer proof of Lemma 1. Note that Lemma 2 is a strengthening of Lemma 1. 2. The Solution This section contains the proof of the key result (Lemma 3) The theorem that Robbins algebras and Boolean ....

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S. Winker. Absorption and idempotency criteria for a problem in near-Boolean algebras. J. Algebra, 153(2):414--423, 1992.

....In terms of the ordering relation on the elements of Robbins algebra, the theorem says that the existence of two elements c and d with d less than or equal to c together with the Robbins axioms is all that is needed to imply Boolean. The theorem was first proved by Winker using induction [12, 13]; McCune later obtained a proof with AC unification. My goal, for years, has been to prove the theorem without induction and without AC unification. In 1996, I made yet another attempt. I assigned heat the value 1 and placed in the (input) hot list only the clause corresponding to the special ....

Winker, S., "Absorption and idempotency criteria for a problem in near-Boolean algebras ", J. Algebra 153, no. 1, 414--423 (December 1992).

....by Argonne s theorem provers are (1) 8x(x x = x) 2) 9c8x(c x = x) and (3) 9c8x(c x = c) Winker then proved (by hand) several weaker conditions sufficient. The two such conditions that play a role in the present work are contained in the following two lemmas. Lemma 1 (S. Winker [16, 17]) A Robbins algebra satisfying 9c9d(c d = c) is a Boolean algebra. Lemma 2 (S. Winker [16, 17] A Robbins algebra satisfying 9c9d(n(c d) n(c) is a Boolean algebra. Appendix B contains a computer proof of Lemma 1. Lemma 2 is a strengthening of Lemma 1 (the hypothesis is weaker) and its ....

....x = c) Winker then proved (by hand) several weaker conditions sufficient. The two such conditions that play a role in the present work are contained in the following two lemmas. Lemma 1 (S. Winker [16, 17] A Robbins algebra satisfying 9c9d(c d = c) is a Boolean algebra. Lemma 2 (S. Winker [16, 17]) A Robbins algebra satisfying 9c9d(n(c d) n(c) is a Boolean algebra. Appendix B contains a computer proof of Lemma 1. Lemma 2 is a strengthening of Lemma 1 (the hypothesis is weaker) and its proof is much more difficult; it is included here mainly for historical purposes (see Sections 2 ....

[Article contains additional citation context not shown here]

S. Winker. Absorption and idempotency criteria for a problem in near-Boolean algebras. J. Algebra, 153(2):414--423, 1992.

....by Argonne s theorem provers are (1) 8x(x x = x) 2) 9c8x(c x = x) and (3) 9c8x(c x = c) Winker then proved (by hand) several weaker conditions sufficient. The two such conditions that play a role in the present work are contained in the following two lemmas. Lemma 1 (S. Winker [15, 16]) A Robbins algebra satisfying 9c9d(c d = c) is a Boolean algebra. Lemma 2 (S. Winker [15, 16] A Robbins algebra satisfying 9c9d(n(c d) n(c) is a Boolean algebra. Appendix B contains a computer proof of Lemma 1. Note that Lemma 2 is a strengthening of Lemma 1. 2. The Solution This ....

....x = c) Winker then proved (by hand) several weaker conditions sufficient. The two such conditions that play a role in the present work are contained in the following two lemmas. Lemma 1 (S. Winker [15, 16] A Robbins algebra satisfying 9c9d(c d = c) is a Boolean algebra. Lemma 2 (S. Winker [15, 16]) A Robbins algebra satisfying 9c9d(n(c d) n(c) is a Boolean algebra. Appendix B contains a computer proof of Lemma 1. Note that Lemma 2 is a strengthening of Lemma 1. 2. The Solution This section contains the the key result the proof of Lemma 3. The theorem that Robbins algebras and ....

[Article contains additional citation context not shown here]

S. Winker. Absorption and idempotency criteria for a problem in near-Boolean algebras. J. Algebra, 153(2):414--423, 1992.

....is a 1 1 map, we could delete the outermost n on both sides, obtaining n(x o n(y) o n(n(x) o n(y) y. which is axiom (H) so the algebra is Boolean. That is, in a non Boolean Robbins algebra, the map n is onto but not 1 1, which implies that A is infinite. A much deeper result is due to Winker [10, 11], who showed that every non Boolean Robbins algebra must satisfy the inequalities 8x8y(x o y 6= x) and 8x8y(n(x o y) 6= n(x) Using this, one may derive other structural properties of these algebras. For example, it is clear that in a non Boolean Robbins algebra, H) must fail somewhere; but ....

S. Winker, Absorption and Idempotency Criteria for a Problem in Near-Boolean Algebras, J. Algebra 153 (1992) 414 -- 423.

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Winker, S., "Absorption and idempotency criteria for a problem in near-Boolean alge-

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