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The concept of linear terms of a given type $\tau$ (terms in which each variable occurs at most once) was introduced by M. Couceiro and E. Lethonen in [Galois theory for sets of operations closed under permutation, cylindrification and composition, Algebr. Univ.67 (2012) 273–297, doi:10.1007/s00012-012-0184-1] (see also [T. Changphas, K. Denecke and B. Pibaljommee, Linear terms and linear hypersubstitutions, SEAMS Bull. Math.40 (2016) (to be published).]). In this paper, we introduce binary partial operations ${+}_g$ and ${*}_g$ on the set $W_{\tau }^{\mathrm{\text{lin}}}(X)$ of all linear terms of type $\tau .$ Then we characterize regular elements and Green’s relations in these partial algebras.

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.

A set $C$ of operations defined on a nonempty set $A$ is said to be a clone if $C$ is closed under composition of operations and contains all projection mappings. The concept of a clone belongs to the algebraic main concepts and has important applications in Computer Science. A clone can also be regarded as a many-sorted algebra where the sorts are the $n$-ary operations defined on set $A$ for all natural numbers $n\ge 1$ and the operations are the so-called superposition operations $S_m^n$ for natural numbers $m,n\ge 1$ and the projection operations as nullary operations. Clones generalize monoids of transformations defined on set $A$ and satisfy three clone axioms. The most important axiom is the superassociative law, a generalization of the associative law. If the superposition operations are partial, i.e. not everywhere defined, instead of the many-sorted clone algebra, one obtains partial many-sorted algebras, the partial clones. Linear terms, linear tree languages or linear formulas form partial clones. In this paper, we give a survey on partial clones and their properties.

The superposition operation $S^{n,A},$$n\ge 1,$$n\in \mathbb{N}$, maps to each $(n+1)$-tuple of $n$-ary operations on a set $A$ an $n$-ary operation on $A$ and satisfies the so-called superassociative law, a generalization of the associative law. The corresponding algebraic structures are Menger algebras of rank $n$. A partial algebra of type $(n+1)$ which satisfies the superassociative law as weak identity is said to be a partial Menger algebra of rank $n$. As a generalization of linear terms we define $r$-terms as terms where each variable occurs at most $r$-times. It will be proved that $n$-ary $r$-terms form partial Menger algebras of rank $n$. In this paper, some algebraic properties of partial Menger algebras such as generating systems, homomorphic images and freeness are investigated. As generalization of hypersubstitutions and linear hypersubstitutions we consider $r$-hypersubstitutions.