Central elements of effect algebras
An element \(a\) of an effect algebra \((E,0,1,\oplus)\) is called principal, if \(b\leq a, c\leq a\) implies \(b\oplus c\leq a\). An element \(a\) of \(E\) is called central, if (1) \(a\) and its orthosupplement \(a'\) are principal, and (2) \(b=(b\wedge a)\oplus(b\wedge a')\) for any \(b\in L\). The author shows that condition (2) on \(a\) may be weakened to (2'): \(b\wedge a = b\wedge a'=0\) implies \(b=0\), and the sets \([0,a]\cap[0,b], [0,a']\cap[0,b]\) have maximal elements for any \(b\in E\). This generalizes the authors result from [Tatra Mt. Math. Publ. 10, 55--62 (1997; Zbl 0915.06003)].
scientific article; zbMATH DE number 2146148