Vol. 21, No. 2, 1967

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Homomorphisms and subalgebras of multialgebras

Harry Eldon Pickett

Vol. 21 (1967), No. 2, 327–342

We shall prove that the homomorphisms and subalgebras of a multialgebra that can be studied naturally through its latticeordered representation as an ordinary algebra are limited to its ideal homomorphisms and Birkhoff subalgebras. However, these form a very limited subclass of the interesting homomorphisms and subalgebras. Of nearly equal importance, for example, are the co-ideal homomorphisms, which arise naturally in (say) groups from left coset decompositions by nonnormal subgroups. To emphasize the special nature of ideal homomorphisms, the class of multiquasigroups is introduced, for which every regular mapping qualifies as a homomorphism. We show in general that ideal (co-ideal) homomorphisms correspond to equivalence relations which we call ideals (co-ideals), and the relationship between ideals, co-ideals, and coset decompositions in multiquasigroups is delineated.

Mathematical Subject Classification
Primary: 08.30
Secondary: 06.00
Received: 9 June 1965
Published: 1 May 1967
Harry Eldon Pickett