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.
