Abstract
The variety of Boolean semirings BIR is the variety generated by the two 2-element semirings. We find a complete set of laws for this variety, and show that it is equivalent to the category of partially Stone spaces. We get a detailed description of the finitely generated free Boolean semirings and a normal form for their elements. From this, a formula is obtained for the free spectrum. This formula relates the free spectra of BIR and DL , the variety of bounded distributive lattices. Finally, we show that the relational degree of BIR is 3.
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