A variety of semigroups is calledhyperassociative if the associative law is satisfied as a hyperi... more A variety of semigroups is calledhyperassociative if the associative law is satisfied as a hyperidentity. We give an equational basis for the variety of all hyperassociative semigroups.
International Journal of Algebra and Computation, 2002
Hypersubstitutions map n-ary operation symbols to n-ary terms. Such mappings can be uniquely exte... more Hypersubstitutions map n-ary operation symbols to n-ary terms. Such mappings can be uniquely extended to mappings defined on the set of all terms. It turns out that the kernels of hypersubstitutions are fully invariant congruence relations on the (absolutely free) term algebra of the considered type. For an arbitrary type τ = (n), n ≥ 1, i.e. if one has only one n-ary operation symbol, we will describe all these congruence relations. The results will be applied to solve the hyperunification problem. Further we will give some generalizations to arbitrary types.
We study a semigroup which represents a semigroup of sets of Boolean functions on a finite set us... more We study a semigroup which represents a semigroup of sets of Boolean functions on a finite set using the concept of transformations with restricted range. For this semigroup, we determine the algebraic structure. In particular, we characterize the (left, right, and two-sided) ideals and the Green’s relations. Moreover, for each of the Green’s relations, we provide the greatest included congruence. MSC: 20M14; 20M18; 20M20
This paper bases on the well-studied semigroup T(X; Y ) of all transfor- mations on X with restri... more This paper bases on the well-studied semigroup T(X; Y ) of all transfor- mations on X with restricted range Y X. We introduce the semi- group TP (X; Y ) of all non-empty subsets of T(X; Y ) under the operation AB := fab : a 2 A;B 2 Bg. We determine the idempotent and regular ele- ments in TP (X; Y ) for the case that jY j = 2. In particular, we characterize the (maximal) regular subsemigroups of TP (X; Y ), the largest semiband, and the (maximal) idempotent subsemigroups of TP (X; Y ).
In the present paper, we study semigroups of endomorphisms on Clifford semigroups with injective ... more In the present paper, we study semigroups of endomorphisms on Clifford semigroups with injective structure homomorphisms, where the semilattice has a least element. We describe such Clifford semigroups having a regular endomorphism monoid. If the endomorphism monoid on the Clifford semigroup is completely regular then the corresponding semilattice has at most two elements. We characterize all Clifford semigroups \(G_{\alpha}\cup G_{\beta}\) (\(\alpha >\beta \)) with an injective structure homomorphism, where \(G_{\alpha}\) has no proper subgroup, such that the endomorphism monoid is completely regular. In particular, we consider the case that the structure homomorphism is bijective.
In this paper, we determine the relative rank of the semigroup T(X; Y ) of all transformations on... more In this paper, we determine the relative rank of the semigroup T(X; Y ) of all transformations on a nite set X with restricted range Y modulo the semigroup of all extensions of the bijections on Y , modulo the idempotent order-preserving transformations in T(X; Y ), and modulo the semigroup of all order-preserving transformations in T(X; Y ).
We extend the study of doppelsemigroups and introduce the notion of an ordered doppelsemigroup. W... more We extend the study of doppelsemigroups and introduce the notion of an ordered doppelsemigroup. We construct the ordered doppelsemigroup of binary relations on an arbitrary set and prove that every ordered doppelsemigroup is isomorphic to some ordered doppelsemigroup of binary relations. In particular, we obtain an analogue of Cayley's theorem for semigroups in the class of doppelsemigroups. We also describe the representations of ordered doppelsemigroups by binary transitive relations.
A node $x$ appearing in an ordered tree is said to be a leftist node if in the subtree with root ... more A node $x$ appearing in an ordered tree is said to be a leftist node if in the subtree with root $x$, the leaf nearest to the root is the leftmost leaf of that subtree. Assuming that all trees belonging to a family of simply generated trees with a specified number of nodes and leaves are equally likely, we present a general approach to the computation of the average number of leftist nodes appearing in such a tree. If the number of nodes in a simply generated tree are specified only, that approach yields exact asymptotical formulae for the average number of leftist nodes, for the average number of internal leftist nodes and for the variances of the corresponding random variables. We illustrate these general results by applying them to various families of simply generated trees, such as $t$-ary trees, binary trees, unary-binary trees, unbalanced 2,3-trees, ordered trees, ordered trees without unary nodes and ordered trees with even node degrees.
In this paper, we determine the relative rank of the semigroup [Formula: see text] of all orienta... more In this paper, we determine the relative rank of the semigroup [Formula: see text] of all orientation-preserving transformations on infinite chains modulo the semigroup [Formula: see text] of all order-preserving transformations.
A variety of semigroups is calledhyperassociative if the associative law is satisfied as a hyperi... more A variety of semigroups is calledhyperassociative if the associative law is satisfied as a hyperidentity. We give an equational basis for the variety of all hyperassociative semigroups.
International Journal of Algebra and Computation, 2002
Hypersubstitutions map n-ary operation symbols to n-ary terms. Such mappings can be uniquely exte... more Hypersubstitutions map n-ary operation symbols to n-ary terms. Such mappings can be uniquely extended to mappings defined on the set of all terms. It turns out that the kernels of hypersubstitutions are fully invariant congruence relations on the (absolutely free) term algebra of the considered type. For an arbitrary type τ = (n), n ≥ 1, i.e. if one has only one n-ary operation symbol, we will describe all these congruence relations. The results will be applied to solve the hyperunification problem. Further we will give some generalizations to arbitrary types.
We study a semigroup which represents a semigroup of sets of Boolean functions on a finite set us... more We study a semigroup which represents a semigroup of sets of Boolean functions on a finite set using the concept of transformations with restricted range. For this semigroup, we determine the algebraic structure. In particular, we characterize the (left, right, and two-sided) ideals and the Green’s relations. Moreover, for each of the Green’s relations, we provide the greatest included congruence. MSC: 20M14; 20M18; 20M20
This paper bases on the well-studied semigroup T(X; Y ) of all transfor- mations on X with restri... more This paper bases on the well-studied semigroup T(X; Y ) of all transfor- mations on X with restricted range Y X. We introduce the semi- group TP (X; Y ) of all non-empty subsets of T(X; Y ) under the operation AB := fab : a 2 A;B 2 Bg. We determine the idempotent and regular ele- ments in TP (X; Y ) for the case that jY j = 2. In particular, we characterize the (maximal) regular subsemigroups of TP (X; Y ), the largest semiband, and the (maximal) idempotent subsemigroups of TP (X; Y ).
In the present paper, we study semigroups of endomorphisms on Clifford semigroups with injective ... more In the present paper, we study semigroups of endomorphisms on Clifford semigroups with injective structure homomorphisms, where the semilattice has a least element. We describe such Clifford semigroups having a regular endomorphism monoid. If the endomorphism monoid on the Clifford semigroup is completely regular then the corresponding semilattice has at most two elements. We characterize all Clifford semigroups \(G_{\alpha}\cup G_{\beta}\) (\(\alpha >\beta \)) with an injective structure homomorphism, where \(G_{\alpha}\) has no proper subgroup, such that the endomorphism monoid is completely regular. In particular, we consider the case that the structure homomorphism is bijective.
In this paper, we determine the relative rank of the semigroup T(X; Y ) of all transformations on... more In this paper, we determine the relative rank of the semigroup T(X; Y ) of all transformations on a nite set X with restricted range Y modulo the semigroup of all extensions of the bijections on Y , modulo the idempotent order-preserving transformations in T(X; Y ), and modulo the semigroup of all order-preserving transformations in T(X; Y ).
We extend the study of doppelsemigroups and introduce the notion of an ordered doppelsemigroup. W... more We extend the study of doppelsemigroups and introduce the notion of an ordered doppelsemigroup. We construct the ordered doppelsemigroup of binary relations on an arbitrary set and prove that every ordered doppelsemigroup is isomorphic to some ordered doppelsemigroup of binary relations. In particular, we obtain an analogue of Cayley's theorem for semigroups in the class of doppelsemigroups. We also describe the representations of ordered doppelsemigroups by binary transitive relations.
A node $x$ appearing in an ordered tree is said to be a leftist node if in the subtree with root ... more A node $x$ appearing in an ordered tree is said to be a leftist node if in the subtree with root $x$, the leaf nearest to the root is the leftmost leaf of that subtree. Assuming that all trees belonging to a family of simply generated trees with a specified number of nodes and leaves are equally likely, we present a general approach to the computation of the average number of leftist nodes appearing in such a tree. If the number of nodes in a simply generated tree are specified only, that approach yields exact asymptotical formulae for the average number of leftist nodes, for the average number of internal leftist nodes and for the variances of the corresponding random variables. We illustrate these general results by applying them to various families of simply generated trees, such as $t$-ary trees, binary trees, unary-binary trees, unbalanced 2,3-trees, ordered trees, ordered trees without unary nodes and ordered trees with even node degrees.
In this paper, we determine the relative rank of the semigroup [Formula: see text] of all orienta... more In this paper, we determine the relative rank of the semigroup [Formula: see text] of all orientation-preserving transformations on infinite chains modulo the semigroup [Formula: see text] of all order-preserving transformations.
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Papers by Jörg Koppitz