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Near heaps

Ian Hawthorn, Tim Stokes (2011)

Commentationes Mathematicae Universitatis Carolinae

On any involuted semigroup ( S , · , ' ) , define the ternary operation [ a b c ] : = a · b ' · c for all a , b , c ∈ S . The resulting ternary algebra ( S , [ ] ) satisfies the para-associativity law [ [ a b c ] d e ] = [ a [ d c b ] e ] = [ a b [ c d e ] ] , which defines the variety of semiheaps. Important subvarieties include generalised heaps, which arise from inverse semigroups, and heaps, which arise from groups. We consider the intermediate variety of near heaps, defined by the additional laws [ a a a ] = a and [ a a b ] = [ b a a ] . Every Clifford semigroup is a near heap when viewed as a semiheap, and we show that the Clifford semigroup...

New proof of a characterization of geodetic graphs

Ladislav Nebeský (2002)

Czechoslovak Mathematical Journal

In [3], the present author used a binary operation as a tool for characterizing geodetic graphs. In this paper a new proof of the main result of the paper cited above is presented. The new proof is shorter and simpler.

Non-associative geometry and discrete structure of spacetime

Alexander I. Nesterov, Lev Vasilʹevich Sabinin (2000)

Commentationes Mathematicae Universitatis Carolinae

A new mathematical theory, non-associative geometry, providing a unified algebraic description of continuous and discrete spacetime, is introduced.

Nonassociative triples in involutory loops and in loops of small order

Aleš Drápal, Jan Hora (2020)

Commentationes Mathematicae Universitatis Carolinae

A loop of order n possesses at least 3 n 2 - 3 n + 1 associative triples. However, no loop of order n > 1 that achieves this bound seems to be known. If the loop is involutory, then it possesses at least 3 n 2 - 2 n associative triples. Involutory loops with 3 n 2 - 2 n associative triples can be obtained by prolongation of certain maximally nonassociative quasigroups whenever n - 1 is a prime greater than or equal to 13 or n - 1 = p 2 k , p an odd prime. For orders n ≤ 9 the minimum number of associative triples is reported for both general and involutory...

Nonsplitting F-quasigroups

Stephen Gagola III (2012)

Commentationes Mathematicae Universitatis Carolinae

T. Kepka, M.K. Kinyon and J.D. Phillips: The structure of F-quasigroups, J. Algebra 317 (2007), no. 2, 435–461 developed a connection between F-quasigroups and NK-loops. Since NK-loops are contained in the variety generated by groups and commutative Moufang loops, a question that arises is whether or not there exists a nonsplit NK-loop and likewise a nonsplit F-quasigroup. Here we prove that there do indeed exist nonsplit F-quasigroups and show that there are exactly four corresponding nonsplit...

Normality, nuclear squares and Osborn identities

Aleš Drápal, Michael Kinyon (2020)

Commentationes Mathematicae Universitatis Carolinae

Let Q be a loop. If S ≤ Q is such that ϕ ( S ) ⊆ S for each standard generator of  Inn Q , then S does not have to be a normal subloop. In an LC loop the left and middle nucleus coincide and form a normal subloop. The identities of Osborn loops are obtained by applying the idea of nuclear identification, and various connections of Osborn loops to Moufang and CC loops are discussed. Every Osborn loop possesses a normal nucleus, and this nucleus coincides with the left, the right and the middle nucleus. Loops that...

Note on analytic Moufang loops

Eugen Paal (2004)

Commentationes Mathematicae Universitatis Carolinae

It is explicitly shown how the Lie algebras can be associated with the analytic Moufang loops. The resulting Lie algebra commutation relations are well known from the theory of alternative algebras.

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