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Otter: The CADE13 Competition Incarnations
 JOURNAL OF AUTOMATED REASONING
, 1997
"... This article discusses the two incarnations of Otter entered in the CADE13 Automated Theorem Proving Competition. Also presented are some historical background, a summary of applications that have led to new results in mathematics and logic, and a general discussion of Otter. ..."
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This article discusses the two incarnations of Otter entered in the CADE13 Automated Theorem Proving Competition. Also presented are some historical background, a summary of applications that have led to new results in mathematics and logic, and a general discussion of Otter.
The structure of conjugacy closed loops
 Trans. Amer. Math. Soc
, 2000
"... Abstract. We study structure theorems for the conjugacy closed (CC) loops, a specific variety of Gloops (loops isomorphic to all their loop isotopes). These theorems give a description all such loops of small order. For example, if p and q are primes, p<q,andq − 1 is not divisible by p, then th ..."
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Abstract. We study structure theorems for the conjugacy closed (CC) loops, a specific variety of Gloops (loops isomorphic to all their loop isotopes). These theorems give a description all such loops of small order. For example, if p and q are primes, p<q,andq − 1 is not divisible by p, then the only CCloop of order pq is the cyclic group of order pq. Foranyprimeq>2, there is exactly one nongroup CCloop in order 2q, and there are exactly three in order q 2. We also derive a number of equations valid in all CCloops. By contrast, every equation valid in all Gloops is valid in all loops. 1.
Alternative Loop Rings
, 1998
"... The right alternative law implies the left alternative law in loop rings of characteristic other than 2. We also exhibit a loop which fails to be a right Bol loop, even though its characteristic 2 loop rings are right alternative. 1 Introduction Throughout this paper, (L; \Delta) always denotes a l ..."
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The right alternative law implies the left alternative law in loop rings of characteristic other than 2. We also exhibit a loop which fails to be a right Bol loop, even though its characteristic 2 loop rings are right alternative. 1 Introduction Throughout this paper, (L; \Delta) always denotes a loop, with identity element e, and (R; +; \Delta) always denotes an associative commutative ring, with identity element 1 6= 0. Then, RL denotes the loop ring constructed from R and L. Elements of RL are represented by finite formal sums of the form P i!n a i x i , where the x i are elements of L and the a i are elements of R. The sum and product operations on RL are defined in the obvious way. Then, 1e is the identity element of RL. See the survey by Goodaire and Milies [2] for more details, background information, and references to the earlier literature. Note that L is embedded into RL via the map x 7! 1x; we usually write the element 1x simply as x. It is now trivial to verify: Remar...
Every diassociative Aloop is Moufang
 Proc. Amer. Math. Soc
"... Abstract. An Aloop is a loop in which every inner mapping is an automorphism. We settle a problem which had been open since 1956 by showing that every diassociative Aloop is Moufang. 1. ..."
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Abstract. An Aloop is a loop in which every inner mapping is an automorphism. We settle a problem which had been open since 1956 by showing that every diassociative Aloop is Moufang. 1.
Quasigroups, Loops, and Associative Laws
 Journal of Algebra
, 1995
"... We study weakenings of associativity which imply that a quasigroup is a loop. In particular, these weakenings include each of Fenyves' "Extra" loop axioms. x1. Introduction. A quasigroup is a system (G; \Delta) such that G is a nonempty set and \Delta is a binary function on G satisf ..."
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We study weakenings of associativity which imply that a quasigroup is a loop. In particular, these weakenings include each of Fenyves' "Extra" loop axioms. x1. Introduction. A quasigroup is a system (G; \Delta) such that G is a nonempty set and \Delta is a binary function on G satisfying 8xz9!y(xy = z) and 8yz9!x(xy = z). A loop is a quasigroup which has an identity element, 1, satisfying 8x(x1 = 1x = x). See the books [1, 2, 9] for background and references to earlier literature. A group is, by definition, an associative loop. As is wellknown, every quasigroup satisfying the associative law has an identity element, and is hence a group. In this paper we consider weakenings of associativity which also imply that a quasigroup is a loop, even though many of these weakenings do not imply the full associative law. For example, consider the four Moufang identities: M1 : (x(yz))x = (xy)(zx) M2 : (xz)(yx) = x((zy)x) N1 : ((xy)z)y = x(y(zy)) N2 : ((yz)y)x = y(z(yx)) As usual, equations wri...
Primary decompositions in varieties of commutative diassociative loops
"... Abstract. The decomposition theorem for torsion abelian groups holds analogously for torsion commutative diassociative loops. With this theorem in mind, we investigate commutative diassociative loops satisfying the additional condition (trivially satisfied in the abelian group case) that all nth pow ..."
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Abstract. The decomposition theorem for torsion abelian groups holds analogously for torsion commutative diassociative loops. With this theorem in mind, we investigate commutative diassociative loops satisfying the additional condition (trivially satisfied in the abelian group case) that all nth powers are central, for a fixed n. For n = 2, we get precisely commutative C loops. For n = 3, a prominent variety is that of commutative Moufang loops. Many analogies between commutative C and Moufang loops have been noted in the literature, often obtained by interchanging the role of the primes 2 and 3. We show that the correct encompassing variety for these two classes of loops is the variety of commutative RIF loops. In particular, when Q is a commutative RIF loop: all squares in Q are Moufang elements, all cubes are C elements, Moufang elements of Q form a normal subloop M0(Q) such that Q/M0(Q) is a C loop of exponent 2 (a Steiner loop), C elements of L form a normal subloop C0(Q) such that Q/C0(Q) is a Moufang loop of exponent 3. Since squares (resp. cubes) are central in commutative C (resp. Moufang) loops, it follows that Q modulo its center is of exponent 6. Returning to the decomposition theorem, we find that every torsion, commutative RIF loop is a direct product of a C 2loop, a Moufang 3loop, and an abelian group with each element of order prime to 6. We also discuss the definition of Moufang elements, and the quasigroups associated with commutative RIF loops. 1.
Experiments with the Hot List Strategy
 Technical Memorandum ANL/MCSTM232, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne
, 1997
"... ..."
Logical Basis for the Automation of Reasoning: Case Studies 1
"... 2.1 Components of the clause language paradigm.................................... 4 ..."
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2.1 Components of the clause language paradigm.................................... 4
A Spectrum of Applications of Automated Reasoning
, 2002
"... The likelihood of an automated reasoning program being of substantial assistance for a wide spectrum of applications rests with the nature of the options and parameters it o ers on which to base needed strategies and methodologies. This article focuses on such a spectrum, featuring W. McCune's ..."
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The likelihood of an automated reasoning program being of substantial assistance for a wide spectrum of applications rests with the nature of the options and parameters it o ers on which to base needed strategies and methodologies. This article focuses on such a spectrum, featuring W. McCune's program OTTER, discussing widely varied successes in answering open questions, and touching on some of the strategies and methodologies that played a key role. The applications include nding a rst proof, discovering single axioms, locating improved axiom systems, and simplifying existing proofs. The last application is directly pertinent to the recently found (by R. Thiele) Hilbert's twentyfourth problemwhich is extremely amenable to attack with the appropriate automated reasoning programa problem concerned with proof simpli cation. The methodologies include those for seeking shorter proofs and for nding proofs that avoid unwanted lemmas or classes of term, a speci c option for seeking proofs with smaller equational or formula complexity, and a di erent option to address the variable richness of a proof. The type of proof one obtains with the use of OTTER is Hilbertstyle
Logical Basis for the Automation of Reasoning: Case Studies
"... Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 The clause language paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..."
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Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 The clause language paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1 Components of the clause language paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Interplay of the components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 A fragment of the history of automated reasoning 19602002 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4 Answering open questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.1 Robbins algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .