this paper. 1. COMBINING OF MANY SORTED SIGNATURES Let S be a many sorted signature. A gate of S is an element of the operation symbols of S

Summary. The aim of this article is to present the definition and some properties of homomorphisms of many sorted algebras. Some auxiliary properties of many sorted functions also have been shown.

this article. Operations on

Summary. This is the continuation of the sequence of articles on trees (see [2], [4], [5]). The main goal is to introduce joining operations on decorated trees corresponding with operations introduced in [5]. We will also introduce the operation of substitution. In the last section we dealt with trees decorated by Cartesian product, i.e. we showed some lemmas on joining operations applied to such trees.

Summary. In the article we deal with sets of trees and functions yielding trees. So, we introduce the sets of all trees, all finite trees and of all trees decorated by elements from some set. Next, the functions and the finite sequences yielding (finite, decorated) trees are introduced. There are shown some convenient but technical lemmas and clusters concerning with those concepts. In the fourth section we deal with trees decorated by Cartesian product and we introduce the concept of a tree called a substitution of structure of some finite tree. Finally, we introduce the operations of joining trees, i.e. for the finite sequence of trees we define the tree which is made by joining the trees from the sequence by common root. For one and two trees there are introduced the same operations.

this paper.

Summary. This article is continuation of an article defining products of many sorted algebras [12]. Some properties of notions such as commute, Frege, Args() are shown in this article. Notions of constant of operations in many sorted algebras and projection of products of family of many sorted algebras are defined. There is also introduced the notion of class of family of many sorted algebras. The main theorem states that product of family of many sorted algebras and product of class of family of many sorted algebras are isomorphic.

this paper. 1. Preliminaries For simplicity, we adopt the following rules: S is a non void non empty many sorted signature, U 1 , U 2 , U 3 are non-empty algebras over S, I is a set, A is a many sorted set indexed by I , and B, C are non-empty many sorted sets indexed by I . The following four propositions are true: (1) For every binary relation R and for all sets X , Y such that X ` Y holds (R#Y ) ffi X = R ffi X: (2) Let A be a set, B, C be non empty sets, f be a function from A into B, g b

Summary. The aim of the article is to prove the fact that if extensions of mappings on generator set are equal then these mappings are equal. The article contains the properties of epimorphisms and monomorphisms between Many Sorted Algebras.

Pure terms (without constants)...