标题： 由所有二阶半环生成的代数系 显示英文标题

标题： On the variety generated by all semirings of order two

摘要： 存在十个不同的一般二元半环，按同构关系表示为\( L_2, R_2, M_2, D_2, N_2, T_2, Z_2, W_2, Z_7 \)，和\( Z_8 \)（参见\cite{bk}）。 其中，\( M_2, D_2, W_2 \)和\( Z_8 \)的乘法约简形成半格，而\( L_2, R_2, M_2, D_2, N_2 \)和\( T_2 \)的加法约简是幂等半格，通常称为\emph{幂等半环}。 2015年，Vechtomov和Petrov \cite{vp}研究了由 \( M_2, D_2, W_2 \)和 \( Z_8 \)生成的簇，并证明该簇是有限基的。 同年，Shao和Ren \cite{srii}考察了由六个幂等半环生成的簇，证明该簇的每个子簇都是有限基的。 本文系统地研究了由所有十个二元半环生成的簇。 我们证明该簇恰好包含480个子簇，每个子簇都是有限基的。 显示英文摘要

摘要： There are ten distinct two-element semirings up to isomorphism, denoted \( L_2, R_2, M_2, D_2, N_2, T_2, Z_2, W_2, Z_7 \), and \( Z_8 \) (see \cite{bk}). Among these, the multiplicative reductions of \( M_2, D_2, W_2 \), and \( Z_8 \) form semilattices, while the additive reductions of \( L_2, R_2, M_2, D_2, N_2 \), and \( T_2 \) are idempotent semilattices, commonly referred to as \emph{idempotent semirings}. In 2015, Vechtomov and Petrov \cite{vp} studied the variety generated by \( M_2, D_2, W_2 \), and \( Z_8 \), proving that it is finitely based. In the same year, Shao and Ren \cite{srii} examined the variety generated by the six idempotent semirings, demonstrating that every subvariety of this variety is finitely based. This paper systematically investigates the variety generated by all ten two-element semirings. We prove that this variety contains exactly 480 subvarieties, each of which is finitely based.