Consider the algebra $B=P(\omega)/_{\mathrm{fin}}$ (the quotient of the power set of natural numbers modulo the ideal of finite sets). Is there an infinite strictly descending chain $\{A_i\mid i\in I\}$ of subalgebras of $B$, in the sense that $A_{i+1}$ is a proper subalgebra of $A_i$? By a proper subalgebra, I mean that there is an embedding of $A_{i+1}$ into $A_i$ but not vice versa.