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Ideals of MV-semirings and MV-algebras
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In this paper, we further develop the ideal theory for MV-semirings. Given an MV-couple
(
A
,
S
)
(A,\ S)
, where
A
A
is an MV-algebra and
S
S
is the MV-semiring associated with
A
A
, we know that the prime spectrum
Spec
(
S
)
\operatorname {Spec}(S)
of
S
S
endowed with the Zariski topology and the prime spectrum
coSpec
(
A
)
\operatorname {coSpec}(A)
of
A
A
endowed with the coZariski topology are homeomorphic; for an arbitrary spectral space this is not true. Here, we are interested in what happens when considering the frame of open subsets of these topological spaces. We obtain the following results:
i)
the set of all radical ideals of an MV-semiring
S
S
is a frame isomorphic to
O
(
Spec
(
S
)
)
\mathcal {O}(\operatorname {Spec}(S))
, i.e., the frame of open sets of
Spec
(
S
)
\operatorname {Spec}(S)
;
ii)
the set
Id
(
A
)
\operatorname {Id}(A)
of all ideals of an MV-algebra
A
A
is a frame isomorphic to
O
(
Spec
(
A
)
)
\mathcal {O}(\operatorname {Spec}(A))
, i.e., the frame of open sets of
Spec
(
A
)
\operatorname {Spec}(A)
. One of our main results is that the frame of open sets of
Max
(
A
)
\operatorname {Max}(A)
,
O
(
Max
(
A
)
)
\mathcal {O}(\operatorname {Max}(A))
is, up to isomorphism, a subframe of
O
(
Spec
(
S
)
)
\mathcal {O}(\operatorname {Spec}(S))
. In particular,
O
(
Max
(
A
)
)
\mathcal {O}(\operatorname {Max}(A))
is isomorphic to the frame of open sets of
Min
(
S
)
\operatorname {Min}(S)
generated by the prime ideals of
A
A
.
Title: Ideals of MV-semirings and MV-algebras
Description:
In this paper, we further develop the ideal theory for MV-semirings.
Given an MV-couple
(
A
,
S
)
(A,\ S)
, where
A
A
is an MV-algebra and
S
S
is the MV-semiring associated with
A
A
, we know that the prime spectrum
Spec
(
S
)
\operatorname {Spec}(S)
of
S
S
endowed with the Zariski topology and the prime spectrum
coSpec
(
A
)
\operatorname {coSpec}(A)
of
A
A
endowed with the coZariski topology are homeomorphic; for an arbitrary spectral space this is not true.
Here, we are interested in what happens when considering the frame of open subsets of these topological spaces.
We obtain the following results:
i)
the set of all radical ideals of an MV-semiring
S
S
is a frame isomorphic to
O
(
Spec
(
S
)
)
\mathcal {O}(\operatorname {Spec}(S))
, i.
e.
, the frame of open sets of
Spec
(
S
)
\operatorname {Spec}(S)
;
ii)
the set
Id
(
A
)
\operatorname {Id}(A)
of all ideals of an MV-algebra
A
A
is a frame isomorphic to
O
(
Spec
(
A
)
)
\mathcal {O}(\operatorname {Spec}(A))
, i.
e.
, the frame of open sets of
Spec
(
A
)
\operatorname {Spec}(A)
.
One of our main results is that the frame of open sets of
Max
(
A
)
\operatorname {Max}(A)
,
O
(
Max
(
A
)
)
\mathcal {O}(\operatorname {Max}(A))
is, up to isomorphism, a subframe of
O
(
Spec
(
S
)
)
\mathcal {O}(\operatorname {Spec}(S))
.
In particular,
O
(
Max
(
A
)
)
\mathcal {O}(\operatorname {Max}(A))
is isomorphic to the frame of open sets of
Min
(
S
)
\operatorname {Min}(S)
generated by the prime ideals of
A
A
.
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