CHARACTERIZATIONS OF DERIVATIONS INDUCED BY ENDOMORPHISMS OF BM-ALGEBRAS

Abstract

This paper characterizes derivations associated with a fixed endomorphism f of a BM-algebra. We consider (l,r)-f-derivations, (r,l)-f-derivations, and left f-derivations. The identity r∧s=r, derived from the BM-algebra axioms, reduces their defining equations to one-sided identities. Every (l,r)-f-derivation has the form ω_f (r)=f(r)*(0*c), where c=ω_f (0). Such a mapping is an (r,l)-f-derivation if and only if c=0*c. Left f-derivations coincide with (r,l)-f-derivations, whereas the left-right and right-left classes need not coincide. For each of the three types, regularity is equivalent to equality with f. We characterize kernel membership and prove that the kernel is a subalgebra if and only if the derivation is regular. We also characterize fixed points and prove that regular derivations have fixed-point subalgebras. Finite examples verify the definitions, distinguish the derivation classes, and show why regularity is needed in the subalgebra statements.