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A note on normal varieties of monounary algebras

Ivan Chajda, Helmut Länger (2002)

Czechoslovak Mathematical Journal

A variety is called normal if no laws of the form s = t are valid in it where s is a variable and t is not a variable. Let L denote the lattice of all varieties of monounary algebras ( A , f ) and let V be a non-trivial non-normal element of L . Then V is of the form M o d ( f n ( x ) = x ) with some n > 0 . It is shown that the smallest normal variety containing V is contained in H S C ( M o d ( f m n ( x ) = x ) ) for every m > 1 where C denotes the operator of forming choice algebras. Moreover, it is proved that the sublattice of L consisting of all normal elements of...

A note on triangular schemes for weak congruences

Ivan Chajda, Branimir Šešelja, Andreja Tepavčević (2005)

Czechoslovak Mathematical Journal

Some geometrical methods, the so called Triangular Schemes and Principles, are introduced and investigated for weak congruences of algebras. They are analogues of the corresponding notions for congruences. Particular versions of Triangular Schemes are equivalent to weak congruence modularity and to weak congruence distributivity. For algebras in congruence permutable varieties, stronger properties—Triangular Principles—are equivalent to weak congruence modularity and distributivity.

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