Residual finiteness and related properties in monounary algebras and their direct products

In this paper we discuss the relationship between direct products of monounary algebras and their components, with respect to the properties of residual finiteness, strong/weak subalgebra separability, and complete separability. For each of these properties $${\mathcal {P}}$$ P , we give a criterion $$\mathcal {C_P}$$ C P such that a monounary algebra $$A$$ A has property $${\mathcal {P}}$$ P if and only if it satisfies $$\mathcal {C_P}$$ C P . We also show that for a direct product $$A\times B$$ A × B of monounary algebras, $$A\times B$$ A × B has property $${\mathcal {P}}$$ P if and only if one of the following is true: either both $$A$$ A and $$B$$ B have property $${\mathcal {P}}$$ P , or at least one of $$A$$ A or $$B$$ B are backwards-bounded, a special property which dominates direct products and which guarantees all $${\mathcal {P}}$$ P hold.

Centralizers of a monounary algebra

The centralizer of an algebra is defined as the set of all transformations of the algebra which commute with all fundamental operations. Further, the second centralizer is the set of all transformations which commute with all elements of the (first) centralizer. The paper characterizes the monounary algebras having the property that the first and the second centralizers coincide.

Polymorphism-Homogeneous Monounary Algebras

An n-ary local polymorphism of a given monounary algebra A is a homomorphism from a finitely generated subalgebra of A

The endomorphism spectrum of a monounary algebra

The endomorphism spectrum specA of an algebra A is defined as the set of all positive integers, which are equal to the number of elements in an endomorphic image of A, for all endomorphisms of A. In this paper we study finite monounary algebras and their endomorphism spectrum. If a finite set S of positive integers is given, one can look for a monounary algebra A with S = specA. We show that for countably many finite sets S, no such A exists. For some sets S, an appropriate A with spec A = S are described.For n ∈ ℍ it is easy to find a monounary algebra A with {1, 2, ..., n} = specA. It will be proved that if i ∈ ℍ, then there exists a monounary algebra A such that specA skips i consecutive (consecutive eleven, consecutive odd, respectively) numbers.Finally, for some types of finite monounary algebras (binary and at least binary trees) A, their spectrum is shown to be complete.

IRREDUCIBLE QUASIORDERS OF MONOUNARY ALGEBRAS

Rooted monounary algebras can be considered as an algebraic counterpart of directed rooted trees. We work towards a characterization of the lattice of compatible quasiorders by describing its join- and meet-irreducible elements. We introduce the limit $\cB _\infty $ of all $d$-dimensional Boolean cubes $\Two ^d$ as a monounary algebra; then the natural order on $\Two ^d$ is meet-irreducible. Our main result is that any completely meet-irreducible quasiorder of a rooted algebra is a homomorphic preimage of the natural partial order (or its inverse) of a suitable subalgebra of $\cB _\infty $. For a partial order, it is known that complete meet-irreducibility means that the corresponding partially ordered structure is subdirectly irreducible. For a rooted monounary algebra it is shown that this property implies that the unary operation has finitely many nontrivial kernel classes and its graph is a binary tree.

Some properties of retract lattices of monounary algebras

Necessary and sufficient conditions for a connected monounary algebra (A, f), under which the lattice R ∅(A, f) of all retracts of (A, f) (together with ∅) is algebraic, are proved. Simultaneously, all connected monounary algebras in which each retract is a union of completely join-irreducible elements of R ∅(A, f) are characterized. Further, there are described all connected monounary algebras (A, f) such that the lattice R ∅(A, f) is complemented. In this case R ∅(A, f) forms a boolean lattice.

On finite retract lattices of monounary algebras

For a monounary algebra (A, f) we denote R ∅(A, f) the system of all retracts (together with the empty set) of (A, f) ordered by inclusion. This system forms a lattice. We prove that if (A, f) is a connected monounary algebra and R ∅(A, f) is finite, then this lattice contains no diamond. Next distributivity of R ∅(A, f) is studied. We find a representation of a certain class of finite distributive lattices as retract lattices of monounary algebras.