Algebra universalis

, 80:47 | Cite as

Nonassociative right hoops

  • Peter JipsenEmail author
  • Michael Kinyon
Part of the following topical collections:
  1. Algebras and Lattices in Hawaii


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.


Right residuated magmas Nonassociative hoops Right quasigroups 

Mathematics Subject Classification

06F05 08B15 20N05 



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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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.MathematicsChapman UniversityOrangeUSA
  2. 2.Department of MathematicsUniversity of DenverDenverUSA

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