### Primitive classes of algebras with unary and nullary operations

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We present an algorithm for constructing the free algebra over a given finite partial algebra in the variety determined by a finite list of equations. The algorithm succeeds whenever the desired free algebra is finite.

We investigate the factor of the groupoid of terms through the largest congruence with a given set among its blocks. The set is supposed to be closed for overterms.

We find an independent base for three-variable equations of posets.

We investigate tournaments that are projective in the variety that they generate, and free algebras over partial tournaments in that variety. We prove that the variety determined by three-variable equations of tournaments is not locally finite. We also construct infinitely many finite, pairwise incomparable simple tournaments.

We find several large classes of equations with the property that every automorphism of the lattice of equational theories of commutative groupoids fixes any equational theory generated by such equations, and every equational theory generated by finitely many such equations is a definable element of the lattice. We conjecture that the lattice has no non-identical automorphisms.

The paper contains two remarks on finite sets of groupoid terms closed under subterms and the application of unifying pairs.

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