Fuzzy M-solid Subpseudovarieties

(1)

Vol. 17, No. 1, 2018, 113-122

ISSN: 2279-087X (P), 2279-0888(online) Published on 11 May 2018

www.researchmathsci.org

DOI: http://dx.doi.org/10.22457/apam.v17n1a13

113

Annals of

Fuzzy M-solid Subpseudovarieties

Atthchai Chada

Department of Mathematics, Faculty of Science and Technology, Rajabhat Mahasarakham University, Mahasarakham-44000, Thailand

E-mail: [email protected]

Received 8 April 2018; accepted 9 May 2018

Abstract. In this paper, we define the concept of fuzzy M-solid subpseudovarieties of a given M-solid pseudovariety and show that the lattice of all fuzzy M-solid subpseudo-varieties is a complete sublattice of the lattice of all fuzzy subpseudosubpseudo-varieties of the same M-solid pseudovariety, where M is a submonoid of hypersubstitutions. Moreover, for submonoids M1, M2 of hypersubstitutions with M1 ⊆ M2 , we show that the lattice of all

fuzzy M2-solid subpseudovarieties is a complete sublattice of the lattice of all fuzzy M1 -solid subpseudovarieties.

Keywords: Pseudovarieties; Fuzzy subalgebras; Fuzzy subpseudovarieties; M-solid pseudovarieties, Fuzzy M-solid subpseudovarieties

AMS Mathematics Subject Classification (2010): 03E72, 06D72, 16Y30, 20N10

1. Introduction

The notion of fuzzy set was first introduced by Zadeh[14] in 1965. The first inspiration application for many algebraic structures was the concept of fuzzy group introduced by Rosenfeld [11]._{The fuzzification was applied to subclass of algebras by many authors}

(2)

114

of all fuzzy subpseudovarieties of the same M-solid pseudo-variety. Moreover, for submonoids M₁, M₂_{of hypersubstitutions with M}₁⊆M₂,we show that the lattice of all fuzzy M2-solid subpseudovarieties is a complete sublattice of the lattice of all fuzzy

M1-solid subpseudovarieties.

2. Preliminaries

Let

τ

=(n_i)_i_∈_Ibe a type of algebras with operation symbols(f_i)_i_∈_I where f_iis an n_i-ary operation. An algebra of type

τ

is an ordered pair

Α

;

(

,

(

)

i I

)

A i

f

A

_∈

=

, where A is a

non-empty set and i I A i

f

)

_∈

(

is a sequence of operations on A indexed by a non-empty index set

I such that to each n_i-ary operation symbol f_i there is a correspondingn_i-ary

opera-tion

f

_iAon A. The set A is called the universe of

Α

. We often write

Α

instead of

Α

;

(

,

(

)

i I

)

A i

f

A

_∈

=

. We denote by Alg(

τ

) the class of all algebras of type

τ

and Algf(

τ

)the class of all finite algebras of type

τ

.The concept of a pseudovariety was first

introduced by Eilenberg and Schutzenberger in [4] as a class of finite algebras of the same type

τ

closed under the formations of homomorphic images (H), subalgebras (S) and finite direct products (Pf). A class V of algebras of type

τ

is called a variety if it is closed under taking of the formations (H), (S) and direct product (P), then its finite part, i.e., the class V fin of all finite algebras contained in V is a pseudovariety. Let K⊆ Alg(

τ

). We denote by H(K) the class of all homomorphic images of algebras in K, by S(K) the class of all subalgebras of algebras in K, by P(K) the class of all direct products of algebras in K and by Pf(K) the class of all finite direct products of algebras in K. Instead of H({

Α

}), S({

Α

}), P({

Α

}) and Pf({

Α

}), we write H(

Α

), S(

Α

), P(

Α

) and Pf(

Α

), respectively for every algebra

Α

in Alg(

τ

). Clearly, Algf(

τ

)and the class

of all one-element algebrasΤ(

τ

)are pseudovarieties which are called trivial pseudo-varieties.

Let

X

n

:

=

{

x

1

,...,

x

n

}

be a finite set of variables, Wτ(Xn) be the set of all n-ary

terms of type

τ

and let ( ); ( ) 1 n

n

X W X

W_τ _τ

∞

=

=U with

X

:

=

{

x

₁

,...,

x

_n

,...

}

be the set of all

terms of type

τ

. Then we denote by

F

_τ

( X

)

the absolutely free algebra;

F

_τ

( X

)

:

=

)

(

);

(

W

X

_n

f

_i _i_∈_I

−

τ with (fi):(t1,...,tni)→ fi(t1,...,tni).

−

An equation of type

τ

is a

pair( ts, )fromW_τ( X)such pairs are commonly written s≈t. An equation s≈t is an identity of algebra

Α

, denote byΑ|=s≈tif sΑ ≈tΑ,wheresΑand tΑare the term operations induced by terms sand t on Α.

(3)

115

I

I i

i

Sub

V

i

W

∈

=

Ι

∈

∧

{

(

)

|

}

and

for every family

{

W

_i

|

i

∈

Ι

}

of subpseudovarieties of V . Let

K

be a class of algebras of

type

τ

, the class HSPf(K) the smallest pseudovariety containing K. Let

A

be a non-empty set and

P

(A

)

be the power set of

A

. A mapping

γ

:

P

(

A

)

→

P

(

A

)

is called a closure operator on A if for any

X

,

Y

∈

P

(

A

)

, the following conditions hold:

1.

X

⊆

γ

(X

)

(extensivity),

2.

X

⊆

Y

⇒

γ

(

X

)

⊆

γ

(

y

)

_{(monotonicity),} 3.

γ

(

γ

(

X

))

⊆

γ

(

X

)

(idempotency).

3. Fuzzy subpseudovarieties

In this section, we present the concept of fuzzy subpseudovarieties of a pseudovariety V and some results which shown in [1].

Definition 3.1. [1] Let V be a pseudovariety of type τ . A mapping

λ

:

V

→

[

0 ,

1 ]

is called fuzzy subpseudovarieties of V if the following conditions hold:

(1.)

∀

Β

∈

V

∀

Α

∈

H

(

Β

)

,

λ

(

Α

)

≥

λ

(

Β

)

,

(2.)

∀

Β

∈

V

∀

Α

∈

S

(

Β

)

,

λ

(

Α

)

≥

λ

(

Β

)

and

(3.)

(

)

inf{

(

)

|

1 }

1

n

i

i i

n

i

≤

Α

≥

Α

∏

=

λ

where

{

Α

_i

|

1 ≤

i

≤

n

}

⊆

Algf(

τ

).

We denote by

FPV

(

τ

)

the set of all fuzzy subpseudovarieties of V . By the definition

above, we note that if

Α

,

Β

∈

V

such that

Α

is isomorphic to

Β

, then

λ

(

Α

)

=

λ

(

Β

)

. Since every one element algebra of type

τ

is contained in every pseudovariety of type

τ

.

A fuzzy subset

λ

of a pseudovariety V of type

τ

is a function

λ

:

V

→

[

0 ,

1 ]

and for

]

1 ,

0 [

∈

t

the set

λ

_t

=

{

Α

∈

V

|

λ

(

Α

)

≥

t

}

is called level subset of V.

Next proposition is correspondence to the definition of fuzzy subpseudovarieties.

Proposition 3.1. [1] Let V be a pseudovariety of type

τ

,

λ

:

V

→

[

0 ,

1 ]

be a fuzzy

subset of V . Then

λ

is a fuzzy subpseudovarieties of V if and only if for all

t

∈

[

0 ,

1 ]

the level subset

λ

_t

≠

φ

is a subpseudovariety of V .

Let V be a pseudovariety of type

τ

and

λ

,

ν

∈

FPV

(

τ

)

. We define the order on

)

(

τ

FPV

by

λ

≤

ν

(sometime written by

λ

⊆

ν

) which

λ

≤

ν

mean

λ

(

Α

)

≤

ν

(

Α

)

for all

Α

∈

V

. We review the notions of unions and intersections of fuzzy subset of V . Let

}

|

{

λ

_i

i

∈

I

be a family of fuzzy subsets of V. Arbitrary intersections and unions are

I

{

(

)

|

U

}

|

)

(

{

W

Sub

V

i

W

Sub

V

W

I i

i

∈

Ι

=

∈

⊆

∨

(4)

116 defined by

}

|

)

(

inf{

:

)

)(

(

Α

=

Α

∈

Ι

∈

i

i I

i

λ

I

and

}

|

)

(

sup{

:

)

)(

(

Α

=

Α

∈

Ι

∈

i

i i

I i

λ

U

.

Then it is easy the verify that arbitrary intersection of fuzzy subpseudovarieties of Vis again a fuzzy subpseudovarieties of V. In general, the union of fuzzy subpseudovarieties of V need not to be a fuzzy subpseudovarieties of V(see in [1]). For a fuzzy subset

λ

of V, we define the fuzzy subpseudovarieties generated by

λ

by

I

{

(

)

|

}

:

ν

τ

λ

ν

λ

=

∈

FPV

⊆

F .

Next, we consider the lattice of all fuzzy subpseudovarieties of V . Let

{

λ

_i

|

i

∈

I

}

be a

family of fuzzy subpseudovarieties of V . We define the meet ∧ and the join ∨ on

)

(

τ

FPV

as follow:

i I i i I i

λ

I

∈ ∈

=

∧

:

and

λ

:

=

I

{

λ

∈

(

τ

)

|

U

λ

⊆

λ

}

∈

∨

I i

i i

I i

FPV

.

Then the following theorem which is easy to verify.

Theorem 3.1. [1] The lattice of all fuzzy subpseudovarieties of V denoted by FPV(

τ

)

,

);

(

:

=

FPV

τ

∧

∨

forms a complete lattice which the least and the greatest elements, say 0, 1, respectively where 0(Α)=0, 1(Α)=1 for all Α∈V.

4. Fuzzy M-solid subpseudovarieties

In this section, we assume that V is an M-solid pseudovariety of type

τ

and then we give the notion of fuzzy M-solid subpseudovarieties. First of all we start with the concept of hypersubstitutions.

A mapping

σ

:

{

f

i

|

i

∈

I

}

→

W

τ

(

X

)

which maps each ni-ary operation symbol to

an ni-ary term is called a hypersubstitution of type

τ

. Each hypersubstitution

σ

induces

a map ^

σ

on the set of all terms which is defined by

(1)

(

x

)

:

=

x

^

σ

if

x

∈

X

is a variable,

(2)

[

(

,...,

)]

:

(

)(

[

],...,

[

])

^ 1 ^ 1

^

i

i i n

n

i

t

f

t

f

σ

=

for composite terms

(

₁

,...,

)

i

n

i

t

f

.

On the set

Hyp

(

τ

)

of all hypersubstitutions of type

τ

, we define a binary operation

)

(

)

(

:

Hyp

τ

Hyp

τ

h

→

o

by 1 ₂

^ 2

1

σ

o

_h

=

o

where

o

is the usual composition of functions. Then together with the identity element

σ

_idmapping each

f

_ito term

)

,...,

(

₁

i

n

i

x

f

, we obtain a monoid

(

Hyp

(

τ

);

o

_h

,

σ

_id

).

Let

Α

=

(

A

;

(

f

_iA

)

_i_∈_I

)

be an

(5)

117

determined by Α and

σ

.

For the class

K

⊆

Algf (

τ

) and for submonoid

)

(

τ

Hyp

M

⊆

we define

}.

,

|

)

(

{

:

]

[

K

M

K

A

M

=

σ

Α

σ

∈

Α

∈

χ

We note that

χ

_MAis a closure operator on Algf(

τ

). A pseudovarity

V

of type

τ

is called

M –solid if

χ

_MA

[

V

]

=

V

.

Now we give the notion of fuzzy M-solid subpseudovarity of a given M-solid pseudovariety.

Definition 4.1. Let

M

be a submonoid of

Hyp

(

τ

)

and let

V

be an M-solid pseudovariety of type

τ

. A fuzzy subset

λ

of

V

is called a fuzzy M-solid subpseudovarity if

(1)

λ

is a fuzzy subpseudovarity of

V

, (2) ∀

σ

∈M ∀Α∈V,

λ

(

σ

(Α))≥

λ

(Α).

We denote by FMP(V)the set of all fuzzy M-solid subpseudovarities of

V

.

Proposition 4.1. Let

V

τ

,

λ

:

V

→

[

0 ,

1 ]

be a fuzzy subset of

V

.

Then

λ

is a fuzzy M-solid subpseudovarity of

V

iff for all

t

∈

[

0 ,

1 ]

the level subset

λ

_t

≠

φ

is an M-solid subpseudovarity of

V

.

Proof: Assume that

λ

V

and let

t

∈

[

0 ,

1 ].

Since

V

is a pseudovariety and by Proposition 3.1, we have

λ

_tis a subpseudovariety of

V

.

Let

M

∈

σ

and

Α

∈

λ

_t

.

By assumption, we have

λ

(

σ

(Α))≥

λ

(Α)≥t. Hence,

σ

(Α)∈

λ

_t. Altogether,

λ

_t is an M-solid subpseudovarity of

V

.

Conversely, assume that for every

t

∈

[

0 ,

1 ],

λ

_t

≠

φ

is an M –solid subpseudovarity

of

V

.

Since

V

is a pseudovariety and by Proposition 3.1, we have

λ

is a fuzzy subpseudovariety of

V

.

Let

σ

∈

M

and

Α

∈

λ

_t

.

We choose t=

λ

(Α)and by assumption, we have

σ

(Α)∈

λ

_t. So

λ

(

σ

(Α))≥t =

λ

(Α). Therefore,

λ

V

.

The following proposition is a consequence of Proposition 4.1.

Proposition 4.2. Let W be a subclass of an M-solid pseudovariety

V

.

Then the characteristic function

λ

_W is a fuzzy M-solid subpseudovarity of

V

iff

W

is an M-solid subpseudovarity of

V

.

For fuzzy subset

µ

of an M-solid pseudovarity of

V

,

we define the fuzzy M-solid subpseudovarity of

V

generated by

µ

by

I

{

(

)

|

}.

:

λ

τ

µ

λ

µ

=

∈

FMPV

⊆

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118

Let

K

be subclass of M-solid pseudovarity of

V

,

the class HSPf

(

[

K

])

A M

χ

is the M -solid subpseudovarity of

V

generated by

K

.

Lemma 4.1. Let

V

be a pseudovariety of type

τ

and

λ

V

.

Define a mapping

ν

:

V

→

[

0 ,

1 ]

by for every

Α

∈

V

,

∈

Α

∈

=

Α

)

:

sup{

[

0 ,

1 ]

|

(

t

ν

HSPf

(

λ

t

)}

.

Then

ν

=

λ

_F

.

Proof: Let

Α

∈

V

and let

I

_Α

=

{t

∈

[

0 ,

1 ]

|

Α

∈

HSPf

(

λ

_t

)}.

First, we prove that

ν

is a

fuzzy subpseudovarity of

V

.

It sufficient to prove that for all

t

∈

Im(

ν

),

ν

_t is an

sub-pseudovarity of

V

.

Let

t

∈

Im(

ν

)

and

1 ,

n

t

_n

=

−

n∈ ℕ

\

{

0 }.

Let

Α

∈

ν

_t

.

Then

,

)

(

Α

≥

t

ν

so

ν

(

Α

)

≥

t

n

,

for all n∈ ℕ

\

{

0 }

. Hence, there exists

s

∈

I

Αsuch that

s

≥

t

n

,

since if for all

s

∈

I

_Α

,

s

≤

t

_n

,

then

ν

(

Α

)

=

sup

I

_Α

≤

t

_n

.

This give a contradiction. Thus,

n

t

s

λ

⊆

and so

Α

∈

HSPf

(

λ

_s

)

⊆

HSPf

(

)

n

t

λ

for all n∈ ℕ

\

{

0 }.

Therefore,

I

} 0 { \ Ν ∈

∈

Α

n

HSPf

(

).

n

t

λ

Conversely, let

I

} 0 \{ Ν ∈

∈

Α

n

HSPf

(

),

n

t

λ

then

t

_n

∈

I

_Α for all n∈

ℕ

\

{

0 }

. Then

=

−

1 ≤

sup

I

Α

=

ν

(

Α

)

n

t

_n for all n∈ ℕ

\

{

0 }

Hence,

ν

(

Α

)

≥

t

,

i.e.,

.

t

ν

∈

Α

Then

I

} 0 { \ Ν ∈

=

n t

ν

HSPf

(

λ

t_n

),

which mean that

ν

tis an subpseudovarity of

V

and

by Proposition 3.1, we have

ν

is a fuzzy subpseudovariety of

V

.

Next step is to show that

.

ν

λ

⊆

Let

Α

∈

V

.

Since

λ

(

Α

)

∈

I

_Α

,

we have

ν

(

Α

)

≥

λ

(

Α

).

This mean that

λ

⊆

ν

.

Finally, we want to prove that for any fuzzy subpseudovariety

δ

of

V

containing

λ

,

we

have

ν

⊆

δ

.

it is clear that for all

t

∈

[

0 ,

1 ],

we have that HSPf

(

λ

t

)

is a subpseudovariety

of

δ

_t

,

since for all

Α

∈

V

and

Α

∈

λ

_timplies that

t

≤

λ

(

Α

)

≥

δ

(

Α

),

i.e.,

Α

∈

δ

_t and so

HSPf

(

λ

_t

)

is a subpseudovariety of

δ

_t

.

Now, we prove that for every

Α

∈

V

,

).

(

)

(

Α

≤

δ

Α

ν

Let

Α

∈

V

and

t

∈

I

_Α

.

Then

Α

∈

HSPf

(

λ

t

)

⊆

δ

t implies

δ

(

Α

)

≥

t

.

Therefore,

ν

(

Α

)

=

sup

I

_Α

≤

δ

(

Α

)

, i.e.,

ν

⊆

δ

.

Lemma 4.2. Let

V

τ

and

µ

V

.

Define a mapping

λ

:

V

→

[

0 ,

1 ]

by for every

Α

∈

V

,

∈

Α

∈

=

Α

)

:

sup{

[

0 ,

1 ]

|

(

t

λ

HSPf

(

χ

_MA

[

µ

_t

])}

.

Then

.

F

µ

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119

Proof: Let

Α

∈

V

and let

I

_Α

=

{t

∈

[

0 ,

1 ]

|

Α

∈

HSPf

(

χ

_MA

[

µ

_t

])}.

First, we prove that

λ

is a fuzzy M-solid subpseudovarity of

V

.

It sufficient to prove that for all

t

∈

Im(

λ

),

λ

_t

is an M-solid subpseudovarity of

V

.

Let

t

∈

Im(

λ

)

and

1 ,

n

t

_n

=

−

n∈ ℕ

\

{

0 }.

Let

.

t

λ

∈

Α

Then

λ

(

Α

)

≥

t

,

so

λ

(

Α

)

≥

t

_n

,

for all n∈ ℕ

\

{

0 }.

Hence, there exists

Α

∈

I

s

such that

s

≥

t

_n

,

since if for all

s

∈

I

_Α

,

s

≤

t

_n

,

then

λ

(

Α

)

=

sup

I

_Α

≤

t

_n

.

This

give a contradiction. Thus,

µ

s

⊆

µ

t_nand so

Α

∈

HSPf

(

[

s

])

⊆

A

M

µ

χ

HSPf

(

[

t_n

])

A

M

µ

χ

for

all n∈ ℕ

\

{

0 }.

since

χ

_MAis a closure operator. Therefore,

I

} 0 \{ Ν ∈

∈

Α

n

HSPf

(

[

]).

n

t A

M

µ

χ

Conversely, let

I

} 0 \{ Ν ∈

∈

Α

n

HSPf

(

χ

_MA

[

µ

_t_n

]),

then

t

_n

∈

I

_Α for all

n

∈

Ν

\

{

0 }.

Then

)

(

sup

1 _≤

₌

_Α

−

=

I

Α

λ

n

t

_n for all n∈ ℕ

\

{

0 }.

Hence,

λ

(

Α

)

≥

t

,

i.e.,

Α

∈

λ

t

.

Then

I

} 0 { \ Ν ∈

=

n t

λ

HSPf

(

[

t_n

]),

A

M

µ

χ

which mean that

λ

tis an M-solid subpseudovarity of

V

and

by Proposition 4.1, we have

λ

is a fuzzy M-solid subpseudovariety of

V

.

Next step is to show that

µ

⊆

λ

.

Let

Α

∈

V

.

Since

µ

(

Α

)

∈

I

_Α

,

we have

λ

(

Α

)

≥

µ

(

Α

).

This mean

that

µ

⊆

λ

.

Finally, we want to prove that for any fuzzy M-solid subpseudovariety

α

of

V

containing

µ

,

we have

λ

⊆

α

.

it is clear that for all

t

∈

[

0 ,

1 ],

we have HSPf

(

[

_t

])

A

M

µ

χ

is an M-solid subpseudovariety of

α

t

,

since for al

Α

∈

V

and

Α

∈

µ

t implies that

),

(

)

(

Α

≥

Α

≤

µ

α

t

i.e.,

Α

∈

α

_t and so HSPf

(

[

_t

])

A

M

µ

χ

is an M-solid subpseudovariety of

α

t

.

Now, we prove that for every

Α

∈

V

,

λ

(

Α

)

≤

α

(

Α

).

Let

Α

∈

V

and

t

∈

I

Α

.

Then

∈

Α

HSPf

(

χ

_MA

[

µ

_t

])

⊆

α

_t implies that

α

(

Α

)

≥

t

.

Therefore,

λ

(

Α

)

=

sup

I

_Α

≤

α

(

Α

),

i.e.,

λ

⊆

α

.

By Theorem 3.1, we obtain the following lemma.

Lemma 4.3. The lattice FMP(V)

:

=

(

FMP

(

V

);

∧

,

∨

)

of all fuzzy M-solid subpseudova-rieties of V is a complete lattice.

As the same result in [1], we obtain that the lattice of all M-solid subpseudovarieties of V can be embedded into the lattice FMP(V).

Lemma 4.4. Let

M

be a submonoid of

Hyp

(

τ

),

V

τ

and

{

λ

_j

|

j

∈

J

}

⊆

FMP

(

V

).

Then t

J j

j

)

(

U

∈

(8)

120 i.e.,

[(

)

]

(

)

t

.

J j j t J j j A

M

U

∈ ∈

=

λ

χ

Proof: Let

{

λ

j

|

j

∈

J

}

⊆

FMP

(

V

).

Since A M

χ

is a closure operator on Algf

(

τ

).

Then

.

)

(

]

)

[(

_t J j j t J j j A

M

U

∈ ∈

⊇

λ

χ

For another inclusion, let t

J j j

)

(

U

∈

Α

λ

and

σ

∈

M

.

We have

.

}

|

)

(

sup{

)

)(

(

_j

j

J

t

J j

j

Α

=

Α

∈

≥

∈

λ

U

Since

λ

_j

(

σ

(

Α

))

≥

λ

_j

(

Α

),

for all

j

∈

J

,

we have

.

}

|

)

(

sup{

}

|

))

(

sup{

λ

_j

σ

Α

j

∈

J

≥

λ

_j

Α

j

∈

J

≥

t

Then

t

J j

j

Α

≥

∈

))

(

)(

(

U

λ

σ

for all

,

M

∈

σ

i.e.,

[(

)

]

(

)

t

.

J j j t J j j A

M

U

∈ ∈

⊆

λ

χ

Therefore,

[(

)

]

(

)

t

.

J j j t J j j A

M

U

∈ ∈

=

λ

χ

Theorem 4.1. The lattice FMP(V)

:

=

(

FMP

(

V

);

∧

,

∨

)

of all fuzzy M-solid

subpseudova-rieties of Vis a complete sublattice of FPV(

τ

)

:

=

(

FPV

(

τ

);

∧

,

∨

)

of all fuzzy subpseu-dovarieties of V.

Proof: Obviously,

FMP

(

V

)

⊆

FPV

(

τ

).

Let

{

λ

_j

|

j

∈

J

}

⊆

FMP

(

V

).

It is clear that if

),

(

τ

λ

_j

FPV

J

j

∧

∈

thenj

∧

∈J

λ

j

∈

FMP

(V

).

Now, we want to show if j

∨

∈J

λ

j

∈

FPV

(

τ

),

then _j

FMP

(V

).

J

j

∨

∈

λ

∈

Let

Α

∈

V

.

By Lemma 4.1 and Lemma 4.4, we obtain

) )(

(∨ Α

∈J j

j

λ

=

sup{t

∈

[

0 ,

1 ]

|

Α

∈

HSPf

}

)

(

_t J j j

U

∈

λ

=sup{t∈[0,1]|Α∈HSPf

(

)

t

}

J j j M

U

∈ Α

_λ

χ

for all

Α

∈

V

.

By Lemma 4.2, we have _j

FMP

(V

).

J

j

∨

∈

λ

∈

This show that the lattice FMP(V) is a

complete sublattice of FPV(

τ

).

Let

V

τ

.It is natural to ask for the relation-ship between the complete lattices FM1P(V)

:

=

(

FM

₁

P

(

V

);

∧

,

∨

)

and FM2P(V)

:

=

)

,

);

(

FM

₂

P

V

∧

∨

when

M

₁and

M

₂are both submonoids of

Hyp

(

τ

),

and

M

₁is a submonoid of

M

₂

.

Theorem 4.2. For any two submonoids

M

1and

M

2of the monoids

Hyp

(

τ

)

with

M

1

⊆

2

M

and

M

₁,

M

₂ solid pseudovariety

V

,

the lattice FM2P(V) is a complete sublattice of the lattice FM1P(V).

Proof: First, we show that

FM

₂

P

(

V

)

⊆

FM

₁

P

(

V

).

Let

λ

∈

FM

₂

P

(

V

),

σ

∈

M

₂and

.

V

∈

(9)

121

.

1

M

∈

σ

Hence,

λ

∈

FM

₁

P

(

V

).

Let

{

λ

j

|

j

∈

J

}

⊆

FM

2

P

(

V

).

It is clear that if

),

(

1

P

V

FM

j J

j

∧

∈

λ

∈

thenj

∧

∈J

λ

j

∈

FM

2

P

(

V

).

Next, we want to show that if

),

(

1

P

V

FM

j J

j

∨

∈

λ

∈

then j

∨

∈J

λ

j

∈

FM

2

P

(

V

).

By Lemma 4.4 and

FM

2

P

(

V

)

⊆

), ( 1PV

FM we have ₂

[(

)

]

(

)

.

₁

[(

)

t

].

J j

j A

M t J j

j t

J j

j A

M

U

∈ ∈

∈

=

λ

χ

λ

χ

Let

Α

∈

V

.

Assume

that j

FM

1

P

(

V

).

J

j

∨

∈

λ

∈

By Lemma 4.2, we have

)

)(

(

∨

Α

∈ j J

j

λ

=sup{t∈[0,1]|Α∈HSPf 1

[(

)

t

]}

J j

j A

M

U

∈

λ

χ

=sup{t∈[0,1]|Α∈ HSPf ₂

[(

)

t

]}.

J

j j A

M

U

∈

λ

χ

This mean that j

FM

2

P

(

V

).

J

j

∨

∈

λ

∈

Therefore, the lattice FM2P(V) is a complete

sublattice of the lattice FM1P(V).

5. Conclusion

We have defined the notion of fuzzy M-solid subpseudovariety of a given M-solid pseudovariety and shown that the lattice of all fuzzy M-solid subpseudovarieties of a given M-solid pseudovariety is a complete sublattice of the lattice of all fuzzy subpseudo-varieties of the M-solid pseudovariety. Moreover, for submonoids M₁, M₂ of hypersub-stitutions with M₁⊆M₂,we show that the lattice of all fuzzy M2-solid

subpseudovarie-ties is a complete sublattice of the lattice of all fuzzy M1-solid subpseudovarieties. It is

known that every M-solid pseudova-rieties can be defined by a set of M-hyperidentities. But the connection between the lattice of all fuzzy M-solid subpseudovarieties of a given M-solid pseudovariety and the lattice of all fuzzy M-hyperidentities of the pseudovariety is still open.

Acknowledgements. The authors would like to thank the referees for valuable suggestions on the paper and thank the Faculty of Science and Technology, Rajabhat Mahasarakham University, Mahasarakham, Thailand for financial support .

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