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nation of the closure of a set in a quasi-algebra (see [2]) by the closure in a certain algebra of a set having one and only one element.

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(1)

BOCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Seria I: PRACE MATEMATYCZNE X (1967)

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE

W. W

aliszew ski

(Łódź)

O n closures o! sets in quasi-algebras

In the present note we consider a problem concerning the determi­

nation of the closure of a set in a quasi-algebra (see [2]) by the closure in a certain algebra of a set having one and only one element.

If / is a function, D om (/) denotes the domain of this function. If Z c D om (/), then f [ Z ] denotes the image of Z under/. If F is a set-valued function, then P F denotes the Cartesian product of all sets F(x), xel)Qm(F). For any set A and any ordinal number Tc, A k denotes the set of all sequences of type Tc having values in A.

Let A be an arbitrary set. By a partial operation defined in the set A we mean an every function / which maps a certain non-empty subset of A k into the set A , where Tc is an ordinal number depending o f / , denoted by m(f) and called the number of members of the partial operation /.

B y a quasi-algebra we mean any ordered pair (A , F } , where A is a set and F is a transfinite sequence of partial operations in A. The quasi- algebra <A , F ) is called an algebra if D om (Ff) = A m(F^ for any feD om (F).

Let = {A , F y be an arbitrary quasi-algebra. For any set M c A let M& denote the smallest of all sets X such that

(1) M <=. X c A

and such that for any ordinal number £eDom(F) (2) F £[ Bom(Fs) r, J “ <Fe>] с X .

The set M& is called the closure of M with respect to л/.

For a given quasi-algebra sć = (A , F } we define == <2A, F *>, where for any feD om (F ) and for Qe(2A)m(Ft\

(3) F*(Q) = F s[Dom(Fs) r, PQ].

Evidently that the quasi - algebra s/* is an algebra, Dom (F) = Dom (F*)

and m{F() = m(F^) for any ^ е Б о т(^ ).

(2)

142 W . W a l i s z e w s k i

Th e o r e m.

For any quasi ^algebra sć

=

F y and for any set M included in A , the identity

(

4

)

holds.

P r o o f . Consider an arbitrary set X satisfying (1) and such that for any £eDom (F) the condition (2) is satisfied. From (1) it follows that

(5) - { i f } c 2x c 2a .

Let £eDom(F). Admit l( = m(Fs) for £eDom(F). Consider an arbi­

trary set ^€F*[(2x )^]. Then there exists a sequence Qe(2x )lt such that 8 = Fę(Q). For any y < l Si the inclusion Qy с X is satisfied. Thus PQ с X 1*. From the conditions (

2

) and (3) we obtain

F*( ( Q ) c X for £€Dom {F).

Then Sc2x . Hence it follows that

(6) FJ[(2X)Z«] с 2X for |eDom(F).

The inclusions (5) and (

6

) yield {M }^ . <= 2X. Therefore (J { if}^ . <= X . Substituting X = we obtain the following inclusion:

(7) U W * * <= JMT-r-

Now, if Г c i f j / , then I d holds for any set satisfying the condition (1) and (2). Let T be an arbitrary set satisfying the conditions:

(

8

) {M } e T c 2

a

and

(9) с T for £eDom{F*).

Denote X — \JT. From (

8

) it follows that the set X satisfies (1).

Note that for any ordinal number к the equality

(10) ( U

2

f = u . p e

Q tTk

holds.

Consider an arbitrary £eDom(F). It follows from (10) (for к = le) and from (3) that

J ; [Dom(_Pf) ^ X 'f] = U F £[Dom(Ft) гч PQ\ = (J ^ ( Q ) =

О.Г1*

o

.

t

'5 ' ;

(3)

Closures o f sets in quasi-algebras 143

From (9) we conclude that the set X satisfies (2) for any |eDom(F).

Consequently, Y c IJ T for any set which satisfies (8) and (9). In partic­

ular, substituting T — { M}^, and Y = 31^, we obtain the inclusion 31 &

cz

(J {31}^, which yields (4) in virtue of (7). This concludes the proof.

References

[1] K . K u r a t o w s k i i A . M o s t o w s k i, Teoria mnogości, Warszawa 1966.

[2] J. S ło m iń s k i , A theory o f extension o f quasi-algebras to algebras, Rozprawy Matematyczne 40 (1964), pp. 1-6 3 .

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