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 P, we give a graphical criterion C P such that a monounary algebra A has property P if and only if it satisfies C P . We also show that for a direct product A × B of monounary algebras, A × B has property P if and only if one of the following is true: either both A and B have property P, or at least one of A or B are backwards-bounded, a special property which dominates direct products and which guarantees all P hold.
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.
A finite unary algebra [Formula: see text] has only countably many countable subdirect powers if and only if every operation [Formula: see text] is either a permutation or a constant mapping.
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