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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 $B$, such that there is an embedding of $A_{i+1}$ into $A_i$, but there is no embedding of $A_i$ into $A_{i+1}$.

2nd EDIT: Klaas Pieter Hart expanded his answer into a paper “Many algebras of $P(\omega)/_{\mathrm{fin}}$” that can be found at arXiv.

1st EDIT: Corrected question according to the remark in the comments.

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 $B$, such that there is an embedding of $A_{i+1}$ into $A_i$, but there is no embedding of $A_i$ into $A_{i+1}$.

2nd EDIT: Klaas Pieter Hart expanded his answer into a paper “Many subalgebras of $P(\omega)/_{\mathrm{fin}}$” that can be found at arXiv.

1st EDIT: Corrected question according to the remark in the comments.

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 $B$, such that there is an embedding of $A_{i+1}$ into $A_i$, but there is no embedding of $A_i$ into $A_{i+1}$.

2nd EDIT: Klaas Pieter Haart expanded his answer into a paper “Many algebras of $P(\omega)/_{\mathrm{fin}}$” that can be found at arXiv.

1st EDIT: Corrected question according to the remark in the comments.

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 $B$, such that there is an embedding of $A_{i+1}$ into $A_i$, but there is no embedding of $A_i$ into $A_{i+1}$.

2nd EDIT: Klaas Pieter Hart expanded his answer into a paper “Many algebras of $P(\omega)/_{\mathrm{fin}}$” that can be found at arXiv.

1st EDIT: Corrected question according to the remark in the comments.

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 $B$, such that there is an embedding of $A_{i+1}$ into $A_i$, but there is no embedding of $A_i$ into $A_{i+1}$.

EDIT: Corrected question according to the remark in the comments.

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 $B$, such that there is an embedding of $A_{i+1}$ into $A_i$, but there is no embedding of $A_i$ into $A_{i+1}$.

2nd EDIT: Klaas Pieter Haart expanded his answer into a paper “Many algebras of $P(\omega)/_{\mathrm{fin}}$” that can be found at arXiv.

1st EDIT: Corrected question according to the remark in the comments.