## พีชคณิตอันดับพรีมอลแบบเทม

Other Title:

Tame order-primal algebras

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Date:

2004

Publisher:

มหาวิทยาลัยศิลปากร

Abstract:

Let h:A ! B be a non-constant homomorphism. It is well-known
that h(A) is a subalgebra of B whose cardinality is greater
than 2. If A is a tame algebra, we give a necessary
and sufficient condition that all images of A under
homomorphisms are tame.
An order-primal algebra A corresponding to an ordered set
(A;?) is an algebra such that T(A) = Pol(A): It was shown in
[3] that if (A;?) is either an antichain or connected,
then A is simple. We can prove that an antichain or a
connected property is a necessary and
sufficient condition for A to be tame. Moreover, we can
prove that every simple algebra is
tame. By these results, we can characterize all order-primal
algebras which are tame; that is,
an order-primal algebra A is tame if and only if (A;?) is
either an antichain or connected.
Furthermore, we can prove that every order-primal algebra
is non-abelian and it has a minimal
set whose cardinality is 2.
In Tame Congruence Theorey, they characterized all
minimal algebras by assigning
type 1-5 and each tame algebra A has at least one type
which is the same type of its
minimal algebra. In [5], they characterize all algebras to
be abelian by using properties
of types of an algebra; that is, an algebra A is abelian if
and only if type A 2 f1; 2g.
By this result, every type of an order-primal algebra is
not type 1 or 2 which implies
that type fAg ? f3; 4; 5g. We can characterize all types
of a tame order-primal algebra
A by the property of the ordered set (A;?) corresponding
to A; that is, (A;?) is
an antichain if and only if type fAg = f3g and (A;?) is
connected if and only if
type fAg = f4g. Moreover, we can find all possible types
of a variety generated by an
order-primal algebra A and give some sufficient conditions
that V (A) is residually small. Let h:A ! B be a non-constant homomorphism. It is well-known
that h(A) is a subalgebra of B whose cardinality is greater
than 2. If A is a tame algebra, we give a necessary
and sufficient condition that all images of A under
homomorphisms are tame.
An order-primal algebra A corresponding to an ordered set
(A;?) is an algebra such that T(A) = Pol(A): It was shown in
[3] that if (A;?) is either an antichain or connected,
then A is simple. We can prove that an antichain or a
connected property is a necessary and
sufficient condition for A to be tame. Moreover, we can
prove that every simple algebra is
tame. By these results, we can characterize all order-primal
algebras which are tame; that is,
an order-primal algebra A is tame if and only if (A;?) is
either an antichain or connected.
Furthermore, we can prove that every order-primal algebra
is non-abelian and it has a minimal
set whose cardinality is 2.
In Tame Congruence Theorey, they characterized all
minimal algebras by assigning
type 1-5 and each tame algebra A has at least one type
which is the same type of its
minimal algebra. In [5], they characterize all algebras to
be abelian by using properties
of types of an algebra; that is, an algebra A is abelian if
and only if type A 2 f1; 2g.
By this result, every type of an order-primal algebra is
not type 1 or 2 which implies
that type fAg ? f3; 4; 5g. We can characterize all types
of a tame order-primal algebra
A by the property of the ordered set (A;?) corresponding
to A; that is, (A;?) is
an antichain if and only if type fAg = f3g and (A;?) is
connected if and only if
type fAg = f4g. Moreover, we can find all possible types
of a variety generated by an
order-primal algebra A and give some sufficient conditions
that V (A) is residually small.

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สาขาวิชาคณิตศาสตร์

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