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Directoids with an antitone involution

Ivan Chajda, Miroslav Kolařík (2007)

Commentationes Mathematicae Universitatis Carolinae

We investigate ⊓ -directoids which are bounded and equipped by a unary operation which is an antitone involution. Hence, a new operation ⊔ can be introduced via De Morgan laws. Basic properties of these algebras are established. On every such an algebra a ring-like structure can be derived whose axioms are similar to that of a generalized boolean quasiring. We introduce a concept of symmetrical difference and prove its basic properties. Finally, we study conditions of direct decomposability of directoids...

F-quasigroups and generalized modules

Tomáš Kepka, Michael K. Kinyon, Jon D. Phillips (2008)

Commentationes Mathematicae Universitatis Carolinae

In Kepka T., Kinyon M.K., Phillips J.D., The structure of F-quasigroups, J. Algebra 317 (2007), 435–461, we showed that every F-quasigroup is linear over a special kind of Moufang loop called an NK-loop. Here we extend this relationship by showing an equivalence between the class of (pointed) F-quasigroups and the class corresponding to a certain notion of generalized module (with noncommutative, nonassociative addition) for an associative ring.

F-quasigroups isotopic to groups

Tomáš Kepka, Michael K. Kinyon, Jon D. Phillips (2010)

Commentationes Mathematicae Universitatis Carolinae

In Kepka T., Kinyon M.K., Phillips J.D., The structure of F-quasigroups, math.GR/0510298, we showed that every loop isotopic to an F-quasigroup is a Moufang loop. Here we characterize, via two simple identities, the class of F-quasigroups which are isotopic to groups. We call these quasigroups FG-quasigroups. We show that FG-quasigroups are linear over groups. We then use this fact to describe their structure. This gives us, for instance, a complete description of the simple FG-quasigroups. Finally,...

Graph Monoids.

F.W. Roush, K.H. Kim, L.G. Makar-Limanov (1982)

Semigroup forum

L’hyperanneau des classes d’adèles

Alain Connes (2011)

Journal de Théorie des Nombres de Bordeaux

J’exposerai ici quelques résultats récents (obtenus en collaboration avec C. Consani [3], [4], [5], [6]) qui portent sur le cas limite de la “caractéristique 1 ”. Le but principal est de montrer que l’espace des classes d’adèles d’un corps global, qui jusqu’à présent n’a été considéré que comme un espace (non-commutatif), admet en fait une structure algébrique naturelle. Nous verrons également que la construction de l’anneau de Witt d’un anneau de caractéristique p > 1 admet un analogue en caractéristique...

Lie ideals in prime Γ-rings with derivations

Nishteman N. Suliman, Abdul-Rahman H. Majeed (2013)

Discussiones Mathematicae - General Algebra and Applications

Let M be a 2 and 3-torsion free prime Γ-ring, d a nonzero derivation on M and U a nonzero Lie ideal of M. In this paper it is proved that U is a central Lie ideal of M if d satisfies one of the following (i) d(U)⊂ Z, (ii) d(U)⊂ U and d²(U)=0, (iii) d(U)⊂ U, d²(U)⊂ Z.

Orthorings

Ivan Chajda, Helmut Länger (2004)

Discussiones Mathematicae - General Algebra and Applications

Certain ring-like structures, so-called orthorings, are introduced which are in a natural one-to-one correspondence with lattices with 0 every principal ideal of which is an ortholattice. This correspondence generalizes the well-known bijection between Boolean rings and Boolean algebras. It turns out that orthorings have nice congruence and ideal properties.

Ring-like operations is pseudocomplemented semilattices

Ivan Chajda, Helmut Länger (2000)

Discussiones Mathematicae - General Algebra and Applications

Ring-like operations are introduced in pseudocomplemented semilattices in such a way that in the case of Boolean pseudocomplemented semilattices one obtains the corresponding Boolean ring operations. Properties of these ring-like operations are derived and a characterization of Boolean pseudocomplemented semilattices in terms of these operations is given. Finally, ideals in the ring-like structures are defined and characterized.

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