We define and examine cardinal invariants connected with algebraic operations on Sierpinski-Zygmund functions.

Recall that a function f: **R**-->**R** is of Sierpinski-Zygmund type
(shortly, an SZ function) if the restriction of f to M is discontinuous for
any set subset M of **R** with cardinality card(M) equal to continuum c,
the cardinality of **R**.

We study the following cardinals, where
$$**R**^{R}
stands for the class of
all functions from **R** to **R**. (Compare [CM], [CR], [Na] and [NR].)

- a(SZ) = min{card(F): F is a subset of
$$
**R**^{R}and there is no h in $$**R**^{R}s.t. for all f in F function h+f is in SZ} - m(SZ) = min{card(F): F is a subset of
$R$
_{0}and there is no h in $$**R**^{R}s.t. for all f in F the product hf is in SZ} - $c$
_{out}(SZ) = min{card(F): F is a subset of $R$_{1}and there is no h in $$**R**^{R}s.t. for all f in F the composition hof is in SZ} - $c$
_{in}(SZ) = min{card(F): F is a subset of $R$_{2}and there is no h in $$**R**^{R}s.t. for all f in F the composition foh is in SZ}

- $R$
_{0}= {f in $$**R**^{R}: card({x: f(x)=0})< c}; - $R$
_{1}= {f in $$**R**^{R}: card({x: f(x)=y})< c for every y in**R**}; - $R$
_{2}= {f in $$**R**^{R}: foh is in SZ for some h in $$**R**^{R}}.

We prove that c<
a(SZ) <
= $2c$
and a(SZ) can be equal to
any regular cardinal between $c+$
and $2c$.
(In particular, each f in $$**R**^{R}
can be expressed as the sum of two SZ
functions.)
Moreover, we compare a(SZ) with a(Darboux), and give the following
combinatorial characterization of a(SZ):

- a(SZ)= min{card(F): F subset of $$
**R**^{R}and for all h in $$**R**^{R}there exists f in F s.t. card({x: f(x)=g(x)})=c}.

- m(SZ)=a(SZ);
- if c is a regular cardinal then c<
$c$
_{out}(SZ)< = $2c$; and - if c=$k+$
for some cardinal k then $c$
_{out}(SZ)=a(SZ); - $c$
_{in}(SZ)=2.

In our considerations we use generalized Martin's Axiom and Lusin sequence axiom.

Bibliography:

- [CM] K. Ciesielski and A. W. Miller,
Cardinal invariants concerning functions whose sum is
almost continuous,
*Real Anal. Exchange 20*(1994--95), 657--673. - [CR] K. Ciesielski and I. Reclaw,
Cardinal invariants
concerning extendable and peripherally continuous functions,
*Real Anal. Exchange 21*(1995-96), 459-472. - [Na] T. Natkaniec, Almost continuity,
*Real Anal. Exchange 17*(1991--92), 462--520. - [NR] T. Natkaniec and I. Reclaw, Cardinal invariants concerning
functions whose product is almost continuous,
*Real Anal. Exchange 20*(1994--95), 281--285.

**DVI and
Postscript files** are available at the
**Topology Atlas**
preprints side.