Algebraic systems, variety of

A class of algebraic systems (cf. Algebraic systems, class of) of a fixed signature $\mathrm{\Omega }$, axiomatizable by identities, i.e. by formulas of the type

$$(\mathrm{\forall }{x}_1)\dots (\mathrm{\forall }{x}_s)P(f_1\dots f_n),$$

where $P$ is some predicate symbol from $\mathrm{\Omega }$ or the equality sign, while $f_1\dots f_n$ are terms of the signature $\mathrm{\Omega }$ in the object variables $x_1\dots x_s$. A variety of algebraic systems is also known as an equational class, or a primitive class. A variety of signature $\mathrm{\Omega }$ can also be defined (Birkhoff's theorem) as a non-empty class of $\mathrm{\Omega }$- systems closed with respect to subsystems, homomorphic images and Cartesian products.

The intersection of all varieties of signature $\mathrm{\Omega }$ which contain a given (not necessarily abstract) class $\mathfrak{K}$ of $\mathrm{\Omega }$- systems is called the equational closure of the class $\mathfrak{K}$( or the variety generated by the class $\mathfrak{K}$), and is denoted by $\mathrm{var}\mathfrak{K}$. In particular, if the class $\mathfrak{K}$ consists of a single $\mathrm{\Omega }$- system $\mathbf{A}$, its equational closure is denoted by $\mathrm{var}\mathbf{A}$. If the system $\mathbf{A}$ is finite, all finitely-generated systems in the variety $\mathrm{var}\mathbf{A}$ are also finite [1], [2].

Let $\mathcal{L}$ be a class of $\mathrm{\Omega }$- systems, let $S\mathcal{L}$ be the class of subsystems of systems of $\mathcal{L}$, let $H\mathcal{L}$ be the class of homomorphic images of the systems from $\mathcal{L}$, and let $\mathrm{\Pi }\mathcal{L}$ be the class of isomorphic copies of Cartesian products of the systems of $\mathcal{L}$. The following relation [1], [2] is valid for an arbitrary non-empty class $\mathfrak{K}$ of $\mathrm{\Omega }$- systems:

$$\mathrm{var}\mathfrak{K}=HS\mathrm{\Pi }\mathfrak{K}.$$

A variety is said to be trivial if the identity $x=y$ is true in each one of its systems. Any non-trivial variety $\mathfrak{M}$ has free systems $F_m(\mathfrak{M})$ of any rank $m$ and $\mathfrak{M}=\mathrm{var}{F}_{{\mathrm{\aleph }}_0}(\mathfrak{M})$[1], [2].

Let $S$ be a set of identities of the signature $\mathrm{\Omega }$ and let $KS$ be the class of all $\mathrm{\Omega }$- systems in which all the identities of $S$ are true. If the equality $\mathfrak{M}=KS$ is satisfied for a variety $\mathfrak{M}$ of signature $\mathrm{\Omega }$, $S$ is known as a basis for $\mathfrak{M}$. A variety $\mathfrak{M}$ is known as finitely baseable if it has a finite basis $S$. For any system $\mathbf{A}$, a basis of the variety $\mathrm{var}\mathbf{A}$ is also known as a basis of identities of the system $\mathbf{A}$. If $\mathfrak{M}$ is a finitely-baseable variety of algebras of a finite signature and if all algebras of $\mathfrak{M}$ have distributive congruence lattices, then each finite algebra $\mathbf{A}$ of $\mathfrak{M}$ has a finite basis of identities [10]. In particular, any finite lattice $⟨\mathbf{A},\vee ,\wedge ⟩$ has a finite basis of identities. Any finite group has a finite basis of identities [3]. On the other hand, there exists a six-element semi-group [5] and a three-element groupoid [6] without a finite basis of identities.

The varieties of $\mathrm{\Omega }$- systems contained in some fixed variety $\mathfrak{M}$ of signature $\mathrm{\Omega }$ constitute under inclusion a complete lattice $L(\mathfrak{M})$ with a zero and a unit, known as the lattice of subvarieties of the variety $\mathfrak{M}$. The zero of this lattice is the variety with the basis $x=y$, $P(x_1\dots x_n)$( $P\in \mathrm{\Omega }$), while its unit is the variety $\mathfrak{M}$. If the variety $\mathfrak{M}$ is non-trivial, the lattice $L(\mathfrak{M})$ is anti-isomorphic to the lattice of all fully-characteristic congruences (cf. Fully-characteristic congruence) of the system $F_{{\mathrm{\aleph }}_0}(\mathfrak{M})$ of countable rank which is free in $\mathfrak{M}$[1]. The lattice $L_{\mathrm{\Omega }}$ of all varieties of signature $\mathrm{\Omega }$ is infinite, except for the case when the set $\mathrm{\Omega }$ is finite and consists of predicate symbols only. The exact value of the cardinality of the infinite lattice $L_{\mathrm{\Omega }}$ is known [1]. The lattice of all lattice varieties is distributive and has the cardinality of the continuum [7], [8]. The lattice of all group varieties is modular, but it is not distributive [3], [4]. The lattice of varieties of commutative semi-groups is not modular [9].

Atoms of the lattice $L_{\mathrm{\Omega }}$ of all varieties of signature $\mathrm{\Omega }$ are known as minimal varieties of signature $\mathrm{\Omega }$. Every variety with a non-unit system contains at least one minimal variety. If the $\mathrm{\Omega }$- system $\mathbf{A}$ is finite and of finite type, then the variety $\mathrm{var}\mathbf{A}$ contains only a finite number of minimal subvarieties [1].

Let $\mathfrak{A},\mathfrak{B}$ be subvarieties of a fixed variety $\mathfrak{M}$ of $\mathrm{\Omega }$- systems. The Mal'tsev product ${\mathfrak{A}}_{\mathfrak{M}}\circ \mathfrak{B}$ denotes the class of those systems $\mathbf{A}$ of $\mathfrak{M}$ with a congruence $\theta$ such that $(\mathbf{A}/\theta )\in \mathfrak{B}$, and all cosets $a/\theta$( $a\in \mathbf{A}$), which are systems in $\mathfrak{M}$, belong to $\mathfrak{A}$. If $\mathfrak{M}$ is the variety of all groups and if $\mathfrak{A}$ and $\mathfrak{B}$ are subvarieties of it, then the product ${\mathfrak{A}}_{\mathfrak{M}}\circ \mathfrak{B}$ is identical with the Neumann product [3]. The product of varieties of semi-groups need not be a variety. A variety $\mathfrak{M}$ of $\mathrm{\Omega }$- systems is called polarized if there exists a term $e(x)$ of the signature $\mathrm{\Omega }$ such that in each system from $\mathfrak{M}$ the identities $e(x)=e(y)$, $F(e(x)\dots e(x))=e(x)$( $F\in \mathrm{\Omega }$) are true. If $\mathfrak{M}$ is a polarized variety of algebras and the congruences in all algebras in $\mathfrak{M}$ are commutable, then the Mal'tsev product ${\mathfrak{A}}_{\mathfrak{M}}\circ \mathfrak{B}$ of any subvarieties $\mathfrak{A}$ and $\mathfrak{B}$ in $\mathfrak{M}$ is a variety. One may speak, in particular, of the groupoid $G_I(\mathfrak{M})$ of subvarieties of an arbitrary variety $\mathfrak{M}$ of groups, rings, etc. If $\mathfrak{M}$ is the variety of all groups or all Lie algebras over a fixed field $P$ of characteristic zero, then $G_I(\mathfrak{M})$ is a free semi-group [1].

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A categorical characterization of varieties of algebraic systems was introduced by F.W. Lawvere [a1]; for a detailed account of this approach see [a2].

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