Rectangular quasigroups and rectangular loops

Kinyon, Michael; Phillips, J. D.

doi:10.1016/j.camwa.2005.01.018

Cited by 2 publications

(3 citation statements)

References 18 publications

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“…We now prove that system (A) axiomatizes the variety of right product quasigroups. It is not difficult to use the results of [5] to prove this, but instead we give a somewhat more enlightening self-contained proof. We start with an easy observation.…”

Section: Axiomsmentioning

confidence: 95%

“…We omit the proof of the equivalence of systems (A) and (B). One can use the results of [5] to prove the system (B) variant of Theorem 2.8 as follows: (A1) and (A2) trivially imply the equations…”

Section: Axiomsmentioning

confidence: 99%

“…By [5], (A3), (A3 ′ ), (B1), (B1 ′ ), (B2 ′ ) and (B2 ′ ) axiomatize the variety of rectangular quasigroups, each of which is a direct product of a left zero semigroup, a quasigroup and a right zero semigroup. By (A1) and (A2), the left zero semigroup factor must be trivial, and so a system satisfying system (B) must be a right product quasigroup.…”

Section: Axiomsmentioning

confidence: 99%

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Right product quasigroups and loops

Phillips¹,

Krapež²,

Kinyon³

2009

Preprint

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Right groups are direct products of right zero semigroups and groups and they play a significant role in the semilattice decomposition theory of semigroups. Right groups can be characterized as associative right quasigroups (magmas in which left translations are bijective). If we do not assume associativity we get right quasigroups which are not necessarily representable as direct products of right zero semigroups and quasigroups. To obtain such a representation, we need stronger assumptions which lead us to the notion of right product quasigroup. If the quasigroup component is a (one-sided) loop, then we have a right product (left, right) loop.We find a system of identities which axiomatizes right product quasigroups, and use this to find axiom systems for right product (left, right) loops; in fact, we can obtain each of the latter by adjoining just one appropriate axiom to the right product quasigroup axiom system.We derive other properties of right product quasigroups and loops, and conclude by showing that the axioms for right product quasigroups are independent.

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Section: Axiomsmentioning

confidence: 95%

Section: Axiomsmentioning

confidence: 99%

Section: Axiomsmentioning

confidence: 99%

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Right product quasigroups and loops

Phillips¹,

Krapež²,

Kinyon³

2009

Preprint

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Automated theorem proving in quasigroup and loop theory

Phillips

Stanovský

2010

AI Communications

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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 state-of-the 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.

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Rectangular quasigroups and rectangular loops

Cited by 2 publications

References 18 publications

Right product quasigroups and loops

Right product quasigroups and loops

Automated theorem proving in quasigroup and loop theory

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