De Morgan algebras are universal

Abstract This paper is concerned with endomorphism monoids of de Morgan algebras. A construction is given of finite ordered sets with an order-inverting involution which are rigid; these correspond under duality to finite rigid de Morgan algebras. It can then be deduced that the variety of de Morgan algebras is universal, and, as a consequence, that for any given monoid M there is a proper class of non-isomorphic de Morgan algebras having M as endomorphism monoid. These conclusions contrast sharply with known results for Boolean algebras, which de Morgan algebras generalize: a Boolean algebra is uniquely determined by its endomorphism monoid.

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