Abstract
The class of nonassociative right hoops, or narhoops for short, is defined as a subclass of naturally ordered right-residuated magmas, and is shown to be a variety. These algebras generalize both right quasigroups and right hoops, and we characterize the subvarieties in which the operation \(x\sqcap y=(x/y)y\) is associative and/or commutative. Narhoops with a left unit are proved to have a top element if and only if \(\sqcap \) is commutative, and their congruences are determined by the equivalence class of the left unit. We also show that the four identities defining narhoops are independent.
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Acknowledgements
This research was supported by the automated theorem prover Prover9 and the finite model builder Mace4, both created by McCune [9]. We would like to thank Bob Veroff for hosting the 2016 Workshop on Automated Deduction and its Applications to Mathematics (ADAM) which is where our collaboration began. We would also like to thank the referee for very useful feedback.
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Dedicated to Ralph Freese, Bill Lampe, and J.B. Nation.
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Presented by W. DeMeo.
This article is part of the topical collection “Algebras and Lattices in Hawaii” edited by W. DeMeo.
Michael Kinyon was partially supported by Simons Foundation Collaboration Grant 359872.
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Jipsen, P., Kinyon, M. Nonassociative right hoops. Algebra Univers. 80, 47 (2019). https://doi.org/10.1007/s00012-019-0618-0
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DOI: https://doi.org/10.1007/s00012-019-0618-0