RESEARCH ON THE THEORY OF FINITE MODELS WITHOUT EQUALITIES(Mathematical Logic and Applications'92)
Libo Lo · Institutional Repositories DataBase (IRDB) · 1993
REPORT SCRIPT) 1) LANGUAGE $L= $ $c_{1},$ $\ldots.,$$c_{m}$ , are constants.$R_{1},$ $\ldots,$ $R_{n}$ are relations with arities $r_{1},$ $\ldots,$ $r_{n}$ .$m,n$ are finite positive integers.No function symbols are in $L$ .No equal signs are used. 2) FORMULASWe use the following formulas:First order formulas.Infinite conjunction of first order formulas.Infinite disjunction of first order formulas.数理解析研究所講究録 第 818 巻 1993 年 35-41 36 3) MODELS $\mathfrak{U}=$ $A$ is the universe of the model.$c_{i}$ is the interpretation of constant $c_{i}$ in $L$ .$R_{i}$ is the interpretation of relation $R_{i}$ in $L$ .$|A|$ is finite.4) EQUIVALENCES ARE DIFFERENT $\mathfrak{U}\models T$ : all sentences of $T$ are satisfied in $\mathfrak{U}$ $T\models T'$ : for all model $\mathfrak{U}\mathfrak{U}\models T\Rightarrow \mathfrak{U}\models T'$ .$T$ is equivalent to $T'$ : For all finite models $T\models T'$ and $T'\models T$ .$T$ is logically equivalent to $T'$ : For all models (finite or infinite) $T\models T'$ and $T'\models T$ .Comparing equivalence and logical equivalence: Let $T$ be the theory of Abelian groups.$\varphi$ : $T\wedge\forall x\exists y(y+y=x)$ .$\psi$ : $T\wedge\forall z(z eq 0arrow z+z eq 0)$ .$\varphi$ and $\psi$ are equivalent but not logically equivalent to each other.