  Several algebraic constructions have obvious definitions and simple properties for BAs: subalgebras, homomorphisms, isomorphisms, and direct products (even of infinitely many algebras). Some other standard algebraic constructions are more peculiar to BAs. An ideal in a BA is a subset \(I\) closed under +, with 0 as a member, and such that if \(a \le b \in I\), then also \(a \in I\). Although not immediately obvious, this is the same as the ring-theoretic concept. There is a dual notion of a filter (with no counterpart in rings in general). A filter is a subset \(F\) closed under \(\cdot\) , having 1 as a member, and such that if \(a \ge b \in F\), then also \(a \in F\). An ultrafilter on \(A\) is a filter \(F\) with the following properties: \(0 \not\in F\), and for any \(a \in A\), either \(a \in F\) or \(-a \in F\). For any \(a \in A\), let