Algebraic systems, quasi-variety of

A class of algebraic systems ( $\mathrm{\Omega }$- systems) axiomatized by special formulas of a first-order logical language, called quasi-identities or conditional identities, of the form

$$(\mathrm{\forall }{x}_1)\dots (\mathrm{\forall }{x}_s)$$

$$[P_1(f_1^{(1)}\dots f_{m_1}^{(1)})\mathrm{\&}\dots \mathrm{\&}{P}_k(f_1^{(k)}\dots f_{m_k}^{(k)})\to$$

$$\to P_0(f_1^{(0)}\dots f_{m_0}^{(0)})],$$

where $P_0\dots P_k\in {\mathrm{\Omega }}_p\cup \{=\}$, and $f_1^{(0)}\dots f_{m_k}^{(k)}$ are terms of the signature $\mathrm{\Omega }$ in the object variables $x_1\dots x_s$. By virtue of Mal'tsev's theorem [1] a quasi-variety $\mathfrak{K}$ of algebraic systems of signature $\mathrm{\Omega }$ can also be defined as an abstract class of $\mathrm{\Omega }$- systems containing the unit $\mathrm{\Omega }$- system $E$, and which is closed with respect to subsystems and filtered products [1], [2]. An axiomatizable class of $\mathrm{\Omega }$- systems is a quasi-variety if and only if it contains the unit $\mathrm{\Omega }$- system $E$ and is closed with respect to subsystems and Cartesian products. If $\mathfrak{K}$ is a quasi-variety of signature $\mathrm{\Omega }$, the subclass ${\mathfrak{K}}_1$ of systems of $\mathfrak{K}$ that are isomorphically imbeddable into suitable systems of some quasi-variety ${\mathfrak{K}}^{\prime }$ with signature ${\mathrm{\Omega }}^{\prime }\supseteq \mathrm{\Omega }$, is itself a quasi-variety. Thus, the class of semi-groups imbeddable into groups is a quasi-variety; the class of associative rings without zero divisors imbeddable into associative skew-fields is also a quasi-variety.

A quasi-variety $\mathfrak{K}$ of signature $\mathrm{\Omega }$ is called finitely definable (or, is said to have a finite basis of quasi-identities) if there exists a finite set $S$ of quasi-identities of $\mathrm{\Omega }$ such that $\mathfrak{K}$ consists of only those $\mathrm{\Omega }$- systems in which all the formulas from the set $S$ are true. For instance, the quasi-variety of all semi-groups with cancellation is defined by the two quasi-identities

$$zx=zy\to x=y,xz=yz\to x=y,$$

and is therefore finitely definable. On the other hand, the quasi-variety of semi-groups imbeddable into groups has no finite basis of quasi-identities [1], [2].

Let $\mathfrak{K}$ be an arbitrary (not necessarily abstract) class of $\mathrm{\Omega }$- systems; the smallest quasi-variety containing $\mathfrak{K}$ is said to be the implicative closure of the class $\mathfrak{K}$. It consists of subsystems of isomorphic copies of filtered products of $\mathrm{\Omega }$- systems of the class $\mathfrak{K}\cup \{E\}$, where $E$ is the unit $\mathrm{\Omega }$- system. If $\mathfrak{K}$ is the implicative closure of a class $\mathfrak{A}$ of $\mathrm{\Omega }$- systems, $\mathfrak{A}$ is called a generating class of the quasi-variety $\mathfrak{K}$. A quasi-variety $\mathfrak{K}$ is generated by one system if and only if for any two systems $\mathbf{A}$, $\mathbf{B}$ of $\mathfrak{K}$ there exists in the class $\mathfrak{K}$ a system $\mathbf{C}$ containing subsystems isomorphic to $\mathbf{A}$ and $\mathbf{B}$[1]. Any quasi-variety $\mathfrak{K}$ containing systems other than one-element systems has free systems of any rank, which are at the same time free systems in the equational closure of the class $\mathfrak{K}$. The quasi-varieties of $\mathrm{\Omega }$- systems contained in some given quasi-variety $\mathfrak{K}$ of signature $\mathrm{\Omega }$ constitute a complete lattice with respect to set-theoretic inclusion. The atoms of the lattice of all quasi-varieties of signature $\mathrm{\Omega }$ are called minimal quasi-varieties of $\mathrm{\Omega }$. A minimal quasi-variety $\mathfrak{M}$ is generated by any one of its non-unit systems. Every quasi-variety with a non-unit system contains at least one minimal quasi-variety. If $\mathfrak{K}$ is a quasi-variety of $\mathrm{\Omega }$- systems of finite signature $\mathrm{\Omega }$, all its sub-quasi-varieties constitute a groupoid with respect to the Mal'tsev $\mathfrak{K}$- multiplication [3].

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In the Western literature, quasi-identities are commonly called Horn sentences (cf. [a1]). For a categorical treatment of quasi-varieties, see [a3]; for their finitary analogue, see [a2]. Mal'tsev's article [a3] may also be found in [a4] as Chapt. 32.

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