Reports on Mathematical Logic

No. 42


On the Variety Generated by Involutive Pocrims

A b s t r a c t. An involutive pocrim (a.k.a.  an $L_0$-algebra) is a residuated integral partially ordered commutative monoid with an involution operator, considered as an algebra. It is proved that the variety generated by all involutive pocrims satisfies no nontrivial idempotent Maltsev condition. That is, no nontrivial $\langle\wedge,\vee,\circ\rangle$-equation holds in the congruence lattices of all involutive pocrims. This strengthens a theorem of A. Wro\'{n}ski. The result survives if we restrict the generating class to totally ordered involutive pocrims.

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