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\begin{center}
{\large{\bf Second Order Varieties of Semirings}} 
 \\[1cm]
{\bf Abstract}
\end{center}

\vspace{1cm}
Hyperidentities in our sense were introduced by W. Taylor [Tay; 81] with the 
aim to extend the concept of identities. Hyperidentities are equations 
consisting of variables and operation symbols in which one can substitute 
for the variables, concrete elements of the appropriate algebraic structure 
and for the operation symbols, concrete term operations of this structure. 
So they are formulas in a second order language where quantification is 
allowed as well for individual variables as for operation symbols. For 
instance one can consider the hyperassociative law as formula of the form:
\[\forall\,x_1\,\forall\,x_2\,\forall\,x_3\,\forall (F(F(x,F(y,z)) \approx 
   F(F(x,y),z)) \]
where for the binary operation symbol $F$ any binary term of the 
considered language can be substituted.

If we request that all identities of a variety are satisfied as 
hyperidentities, we call this variety solid. Solid varieties are second 
order varieties. The important fact that all solid varieties of a given type 
$\tau$ form a complete sublattice of the lattice of all varieties of type 
$\tau$ is a helpful tool to study lattices of varieties.

In our paper, we will apply the general theory of hyperidentities to 
the concrete case of semirings i. e. algebraic structures with two binary 
associative operations where both distributive laws are valid.\\[3mm]


[Tay;81]  Taylor, W., Hyperidentities and hypervarieties, 
Aequationes Mathematicae {\bf 23} (1981), 111-127  


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