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We link the recent theory of $L$-algebras to previous notions of UniversalAlgebra and Categorical Algebra concerning subtractive varieties, commutators,multiplicative lattices, and their spectra. We show that the category of$L$-algebras is subtractive and normal in the sense of Zurab Janelidze, butneither the category of $L$-algebras nor that of pre-$L$-algebras are Mal'tsevcategories, hence in particular they are not semi-abelian. Therefore$L$-algebras are a rather peculiar example of an algebraic structure.

Ideals and Congruences in L-algebras and Pre-L-algebras