-systems of unary algebras. II: Maximal and minimal subalgebras of the direct products of unary algebras
Remark on partially ordered sets, universal algebras and semigroups
Representation of connected monounary algebras by means of irreducibles
The aim of the present paper is to describe all connected monounary algebras for which there exists a representation by means of connected monounary algebras which are retract irreducible in the class (or in ).
Representation of ordered classes by classes of connected unary algebras
Retract irreducibility of connected monounary algebras. I
Retract irreducibility of connected monounary algebras. II
Retract irreducibility of monounary algebras
Retract varieties of monounary algebras
Retracts of monounary algebras corresponding to groupoids
Some monounary algebras with EKP
An algebra is said to have the endomorphism kernel property (EKP) if every congruence on is the kernel of some endomorphism of . Three classes of monounary algebras are dealt with. For these classes, all monounary algebras with EKP are described.
Strong endomorphism kernel property for monounary algebras
All monounary algebras which have strong endomorphism kernel property are described.
Strong retracts of unary algebras
This paper introduces the notion of a strong retract of an algebra and then focuses on strong retracts of unary algebras. We characterize subuniverses of a unary algebra which are carriers of its strong retracts. This characterization enables us to describe the poset of strong retracts of a unary algebra under inclusion. Since this poset is not necessarily a lattice, we give a necessary and sufficient condition for the poset to be a lattice, as well as the full description of the poset.
Subalgebra extensions of partial monounary algebras
For a subalgebra of a partial monounary algebra we define the quotient partial monounary algebra . Let , be partial monounary algebras. In this paper we give a construction of all partial monounary algebras such that is a subalgebra of and .
Subdirect products of certain varieties of unary algebras
J. Płonka in [12] noted that one could expect that the regularization of a variety of unary algebras is a subdirect product of and the variety of all discrete algebras (unary semilattices), but is not the case. The purpose of this note is to show that his expectation is fulfilled for those and only those irregular varieties which are contained in the generalized variety of the so-called trap-directable algebras.
Systems of unary algebras with common endomorphisms. I
Systems of unary algebras with common endomorphisms. II
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 .
The automorphism group of unary algebras.
The category of connected partial unary algebras
The ideal structure of simple ternary algebras