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.

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