Translations in Idempotent Groupoids
Congruence relations on direct products of lattices
On - associated comonotone functions
We give a positive answer to two open problems stated by Boczek and Kaluszka in their paper [1]. The first one deals with an algebraic characterization of comonotonicity. We show that the class of binary operations solving this problem contains any strictly monotone right-continuous operation. More precisely, the comonotonicity of functions is equivalent not only to -associatedness of functions (as proved by Boczek and Kaluszka), but also to their -associatedness with being an arbitrary strictly...
Cardinality of retracts of monounary algebras
For an uncountable monounary algebra with cardinality it is proved that has exactly retracts. The case when is countable is also dealt with.
Test elements and the Retract Theorem for monounary algebras
The term “Retract Theorem” has been applied in literature in connection with group theory. In the present paper we prove that the Retract Theorem is valid (i) for each finite structure, and (ii) for each monounary algebra. On the other hand, we show that this theorem fails to be valid, in general, for algebras of the form , where each is unary and .
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