Abstract
AbstractIn 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.
Publisher
Springer Science and Business Media LLC
Subject
Algebra and Number Theory
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