implicational class

implicational class

In this entry, we extend the notion of an

equational class

(or a

variety

) to a more general notion known as an

implicational class

(or a quasivariety). Recall that an equational class

K

is a class of

algebraic systems

satisfying a set

Σ

of “

equations

” and that

K

is the smallest class satisfying

Σ

. Typical examples are the varieties of groups, rings, or lattices.

An implicational class, loosely speaking, is the

smallest

class of algebraic systems satisfying a set of “

implications

”, where an implication has the form

P

→

Q

, where

P

and

Q

are some

sentences

. Formally, we define an

equational implication

in an algebraic system to be a sentence of the form

(

∀

x

1

)

⋯

(

∀

x

n

)

(

e

1

∧

⋯

∧

e

p

→

e

q

)

,

where each

e

i

is an

identity

of the form

f

i

⁢

(

x

1

,

…

,

x

n

)

=

g

i

⁢

(

x

1

,

…

,

x

n

)

for some

n

-ary

polynomials

f

i

and

g

i

, and

i

=

1

,

…

,

p

,

q

.

Definition

. A class

K

of algebraic systems of the same type (

signature

) is called an

implicational class

if there is a set

Σ

of equational implications such that

K

=

{

A

is a structure

∣

A

is a model in

Σ

}

=

{

A

∣

(

∀

q

∈

Σ

)

→

(

A

⊧

q

)

}

.

Examples

1.

Any equational class is implicational. Each identity

p

=

q

can be thought of as an equational implication

(

p

=

p

)

→

(

p

=

q

)

. In other words, every

algebra

satisfying the identity also

satisfies

the corresponding equational implication, and vice versa.

2.

The class of all

Dedekind-finite

rings. In

addition

to satisfying the identities for being a (unital) ring, each ring also satisfies the equational implication

(

∀

x

)

(

∀

y

)

(

x

y

=

1

)

→

(

y

x

=

1

)

.

3.

The class of all

cancellation semigroups

. In addition to satisfying the identities for being a

semigroup

, each semigroup also satisfies the implications

(

∀

x

)

(

∀

y

)

(

∀

z

)

(

x

y

=

x

z

)

→

(

y

=

z

)

and

(

∀

x

)

(

∀

y

)

(

∀

z

)

(

y

x

=

z

x

)

→

(

y

=

z

)

.

4.

The class

K

of all

torsion free

abelian groups

. In addition to satisfying the identities for being abelian groups, each group also satisfies the set of all implications

{

∀

x

(

n

x

=

0

)

→

(

x

=

0

)

∣

n

is a positive integer

}

.

There is an

equivalent

formulation of an implicational class. Again, let

K

be a class of algebraic systems of the same type (signature)

τ

. Define the following four “

operations

” on

the

classes of algebraic systems of type

τ

:

1.

I

⁢

(

K

)

is the class of all

isomorphic

copies of algebras in

K

,

2.

S

⁢

(

K

)

is the class of all

subalgebras

of algebras in

K

,

3.

P

⁢

(

K

)

is the class of all

product

of algebras in

K

(including the empty products, which means

P

⁢

(

K

)

includes the trivial algebra), and

4.

U

⁢

(

K

)

is the class of all

ultraproducts

of algebras in

K

.

Suppose

X

is any one of the operations above, we say that

K

is

closed

under operation

X

if

X

⁢

(

K

)

⊆

K

.

Definition

.

K

is said to be an

algebraic class

if

K

is

closed under

I

, and

K

is said to be a

quasivariety

if it is algebraic and is closed under

S

,

P

,

U

.

It can be shown that a class

K

of algebraic systems of the same type is implicational iff it is a quasivariety. Therefore, we may use the two terms interchangeably.

As we have seen earlier, a variety is a quasivariety. However, the

converse

is not true, as can be readily seen in the last example above, since a

homomorphic image

of a torsion free

abelian

is in general not torsion free: the homomorphic image of

ϕ

:

ℤ

→

ℤ

n

is a

subgroup

of

ℤ

n

, hence not torsion free.

Title

implicational class

Canonical name

ImplicationalClass

Date of creation

2013-03-22 17:31:35

Last modified on

2013-03-22 17:31:35

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Definition

Classification

msc 08C15

Classification

msc 03C05

Synonym

quasivariety

Synonym

quasiprimitive class

Defines

algebraic class

Defines

equational implication

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