Preliminary on notation: in the question,

• it seems advisable to change $C=T_{k}M$  to $C=T_{k}(M)$ , and change $M=T^{-1}_{k}C$  to $M={T_k}^{-1}(C)$
• it is necessary to change $T_{k}T^{-1}_{k}=I$  to ${T_k}^{-1}\circ T_k=I$ , meaning that the function obtained by applying $T_k$ then its inverse ${T_k}^{-1}$ is identity, with $\forall M, {T_k}^{-1}(T_k(M))=M$.

The question separates the notion of key $K$ from the encryption transformation $T_k$ for this key ($k$ became lowercase and seemingly is now an index, perhaps from a space smaller than the space of all $K$). I would thus say that it could be extended to cover some forms of deterministic asymmetric encryption (like raw/textbook RSA); but that studying the asymmetry and security of such encryption would require additional considerations about the feasibility of computing $T_k$ for arbitrary input knowing $K$ (which would be the public key), when ${T_k}^{-1}$ would only be computable knowing $K'$, with $K'$ computationally impossible to derive from $K$.

As is, the definition only covers that kind of encryption where the output is a function of key and plaintext (only); when encryption often has an additional random input to the encryption function. That's the case in AES-CTR, AES-CBC, and any form of asymmetric encryption that is secure for small messages (like name of a particular students on a public call roll), including secure forms of RSA, and ElGamal encryption. The notation in the question can thus not cover any form of semantically secure asymmetric encryption; which is a huge gap.

Preliminary on notation: in the question,

• it seems advisable to change $C=T_{k}M$  to $C=T_{k}(M)$ , and change $M=T^{-1}_{k}C$  to $M={T_k}^{-1}(C)$
• it is necessary to change $T_{k}T^{-1}_{k}=I$  to ${T_k}^{-1}\circ T_k=I$ , meaning that the function obtained by applying $T_k$ then its inverse ${T_k}^{-1}$ is identity, with $\forall M, {T_k}^{-1}(T_k(M))=M$.

The question separates the notion of key $K$ from the encryption transformation $T_k$ for this key ($k$ became lowercase and seemingly is now an index, perhaps from a space smaller than the space of all $K$). I would say that it could be extended to cover some forms of deterministic asymmetric encryption (like raw/textbook RSA); but that studying the asymmetry and security of such encryption would require additional considerations about the feasibility of computing $T_k$ for arbitrary input knowing $K$ (which would be the public key), when ${T_k}^{-1}$ would only be computable knowing $K'$, with $K'$ computationally impossible to derive from $K$.

As is, the definition only covers that kind of encryption where the output is a function of key and plaintext (only); when encryption often has an additional random input to the encryption function. That's the case in 3DES-CBC or AES-CTR using random IV, and any form of asymmetric encryption that is secure for small messages (like name of a particular students on a public call roll), including secure forms of RSA, and ElGamal encryption. As is, the notation in the question (modeling encryption for a given key as a function of plaintext only) can thus not cover any form of semantically secure asymmetric encryption; which is a huge gap.

Preliminary on notation: in the question,

• it seems advisable to change $C=T_{k}M$  to $C=T_{k}(M)$ , and change $M=T^{-1}_{k}C$  to $M={T_k}^{-1}(C)$
• it is necessary to change $T_{k}T^{-1}_{k}=I$  to ${T_k}^{-1}\circ T_k=I$ , meaning that the function obtained by applying $T_k$ then its inverse ${T_k}^{-1}$ is identity, with $\forall M, {T_k}^{-1}(T_k(M))=M$.

As is, the definition only covers that kind of encryption where the output is a function of key and plaintext only; when encryption often has an additional random input to the encryption function. That's the case in AES-CTR, AES-CBC, and any form of asymmetric encryption that is secure for small messages (like name of a particular students on the call roll), including secure forms of RSA, and ElGamal encryption. The notation in the question can thus not cover any form of semantically secure asymmetric encryption; which is a huge gap.

The notation in the question seems to separate the notion of key $K$ from the encryption transformation $T_k$ for this key ($k$ became lowercase and seemingly is now an index, perhaps from a space smaller than the space of all $K$). I would say that it could be extended to cover some forms of deterministic asymmetric encryption (like textbook RSA); but that studying the asymmetry and security of such encryption would require additional considerations about the feasibility of computing $T_k$ for arbitrary input knowing $K$ (which would be the public key), when ${T_k}^{-1}$ would only be computable knowing $K'$, with $K'$ computationally impossible to derive from $K$.

Preliminary on notation: in the question,

• it seems advisable to change $C=T_{k}M$  to $C=T_{k}(M)$ , and change $M=T^{-1}_{k}C$  to $M={T_k}^{-1}(C)$
• it is necessary to change $T_{k}T^{-1}_{k}=I$  to ${T_k}^{-1}\circ T_k=I$ , meaning that the function obtained by applying $T_k$ then its inverse ${T_k}^{-1}$ is identity, with $\forall M, {T_k}^{-1}(T_k(M))=M$.

The question separates the notion of key $K$ from the encryption transformation $T_k$ for this key ($k$ became lowercase and seemingly is now an index, perhaps from a space smaller than the space of all $K$). I would thus say that it could be extended to cover some forms of deterministic asymmetric encryption (like raw/textbook RSA); but that studying the asymmetry and security of such encryption would require additional considerations about the feasibility of computing $T_k$ for arbitrary input knowing $K$ (which would be the public key), when ${T_k}^{-1}$ would only be computable knowing $K'$, with $K'$ computationally impossible to derive from $K$.

As is, the definition only covers that kind of encryption where the output is a function of key and plaintext (only); when encryption often has an additional random input to the encryption function. That's the case in AES-CTR, AES-CBC, and any form of asymmetric encryption that is secure for small messages (like name of a particular students on a public call roll), including secure forms of RSA, and ElGamal encryption. The notation in the question can thus not cover any form of semantically secure asymmetric encryption; which is a huge gap.