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Graph algebras establish a connection between directed graphs without multiple edges and special universal algebras of type (2,0). We say that a graph G satisfies a term equation s ≈ t if the corresponding graph algebra A(G) satisfies s ≈ t. A class of graph algebras V is called a graph variety if V = Mod_gΣ where Σ is a subset of T(X) × T(X). A graph variety V' = Mod_gΣ' is called a biregular leftmost graph variety if Σ' is a set of biregular leftmost term equations. A term equation s ≈ t is called an identity in a variety V if G satisfies s ≈ t for all G ∈ V.

In this paper we characterize identities in each biregular leftmost graph variety. For identities, varieties and other basic concepts of universal algebra see e.g. [1].

In the present paper, we classify varieties of algebraic systems of the type $((n),(m))$, for natural numbers $n$ and $m$, which are closed under particular derived algebraic systems. If we replace in an algebraic system the $n$-ary operation by an $n$-ary term operation and the $m$-ary relation by the $m$-ary relation generated by an $m$-ary formula, we obtain a new algebraic system of the same type, which we call derived algebraic system. We shall restrict the replacement to so-called “linear” terms and atomic “linear” formulas, respectively.

Graph algebras establish a connection between directed graphs without multiple edges and special universal algebras of type $(2,0)$. We say that a graph $G$ satisfies a term equation $s\approx t$ if the corresponding graph algebra $\underline{A(G)}$ satisfies $s\approx t$. The set of all term equations $s\approx t$, which the graph $G$ satisfies, is denoted by $\mathrm{\text{Id}}(\{G\})$. The class of all graph algebras satisfy all term equations in $\mathrm{\text{Id}}(\{G\})$ is called the graph variety generated by $G$ denoted by ${{𝒱}}_g(\{G\})$. A term is called a linear term if each variable which occurs in the term, occurs only once. A term equation $s\approx t$ is called a linear term equation if $s$ and $t$ are linear terms. This paper is devoted to a thorough investigation of graph varieties defined by linear term equations. In particular, we give a complete description of rooted graphs generating a graph variety described by linear term equations.

We have shown that a variety of left(right) normal bands is closed in some homotypical varieties. Further, we partially generalize a result of Isbell from the class of commutative semigroups to some generalized classes of commutative semigroups by showing that dominion of such semigroups belongs to same classes.