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18
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.
The varieties of quasigroups of BolMoufang type: an equational reasoning approach
 Journal of Algebra
"... Abstract. A quasigroup identity is of BolMoufang type if two of its three variables occur once on each side, the third variable occurs twice on each side, the order in which the variables appear on both sides is the same, and the only binary operation used is the multiplication, viz. ((xy)x)z = x(y ..."
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Abstract. A quasigroup identity is of BolMoufang type if two of its three variables occur once on each side, the third variable occurs twice on each side, the order in which the variables appear on both sides is the same, and the only binary operation used is the multiplication, viz. ((xy)x)z = x(y(xz)). Many wellknown varieties of quasigroups are of BolMoufang type. We show that there are exactly 26 such varieties, determine all inclusions between them, and provide all necessary counterexamples. We also determine which of these varieties consist of loops or onesided loops, and fully describe the varieties of commutative quasigroups of BolMoufang type. Some of the proofs are computergenerated. 1.
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.
Automated theorem proving in quasigroup and loop theory
 NORTHERN MICHIGAN UNIVERSITY, MARQUETTE, MI 49855 USA
"... We survey all known results in the area of quasigroup and loop theory to have been obtained with the assistance of automated theorem provers. We provide both informal and formal descriptions of selected problems, and compare the performance of selected stateofthe art first order theorem provers on ..."
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Cited by 5 (2 self)
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We survey all known results in the area of quasigroup and loop theory to have been obtained with the assistance of automated theorem provers. We provide both informal and formal descriptions of selected problems, and compare the performance of selected stateofthe art first order theorem provers on them. Our analysis yields some surprising results, e.g., the theorem prover most often used by loop theorists does not necessarily yield the best performance.
Automated theorem proving in loop theory
 proceedings of the ESARM workshop
, 2008
"... In this paper we compare the performance of various automated theorem provers on nearly all of the theorems in loop theory known to have been obtained with the assistance of automated theorem provers. Our analysis yields some surprising results, e.g., the theorem prover most often used by loop theor ..."
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In this paper we compare the performance of various automated theorem provers on nearly all of the theorems in loop theory known to have been obtained with the assistance of automated theorem provers. Our analysis yields some surprising results, e.g., the theorem prover most often used by loop theorists doesn’t necessarily yield the best performance. 1
On the universality of some Smarandache loops of Bol Moufang type
 Scientia Magna
"... A Smarandache quasigroup(loop) is shown to be universal if all its f,gprincipal isotopes are Smarandache f,gprincipal isotopes. Also, weak Smarandache loops of BolMoufang type such as Smarandache: left(right) Bol, Moufang and extra loops are shown to be universal if all their f,gprincipal isotop ..."
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A Smarandache quasigroup(loop) is shown to be universal if all its f,gprincipal isotopes are Smarandache f,gprincipal isotopes. Also, weak Smarandache loops of BolMoufang type such as Smarandache: left(right) Bol, Moufang and extra loops are shown to be universal if all their f,gprincipal isotopes are Smarandache f,gprincipal isotopes. Conversely, it is shown that if these weak Smarandache loops of BolMoufang type are universal, then some autotopisms are true in the weak Smarandache subloops of the weak Smarandache loops of BolMoufang type relative to some Smarandache elements. Futhermore, a Smarandache left(right) inverse property loop in which all its f,gprincipal isotopes are Smarandache f,gprincipal isotopes is shown to be universal if and only if it is a Smarandache left(right) Bol loop in which all its f,gprincipal isotopes are Smarandache f,gprincipal isotopes. Also, it is established that a Smarandache inverse property loop in which all its f,gprincipal isotopes are Smarandache f,gprincipal isotopes is universal if and only if it is a Smarandache Moufang loop in which all its f,gprincipal isotopes are Smarandache f,gprincipal isotopes. Hence, some of the autotopisms earlier mentioned are found to be true in the Smarandache subloops of universal Smarandache: left(right) inverse property loops and inverse property loops.
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...
P.: Loops with abelian inner mapping groups: An application of automated deduction
 McCune Festschrift, Lecture Notes in Artificial Intelligence
, 2013
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