@article{DimitrovaKoppitz2020,
  author    = {Dimitrova, Ilinka and Koppitz, J{\"o}rg},
  title     = {On relative ranks of the semigroup of orientation-preserving transformations on infinite chains},
  series = {Asian-European journal of mathematics},
  volume    = {14},
  journal   = {Asian-European journal of mathematics},
  number    = {08},
  publisher = {World Scientific},
  address   = {Singapore},
  issn      = {1793-5571},
  doi       = {10.1142/S1793557121501461},
  pages     = {15},
  year      = {2020},
  abstract  = {In this paper, we determine the relative rank of the semigroup OP(X) of all orientation-preserving transformations on infinite chains modulo the semigroup O(X) of all order-preserving transformations.},
  language  = {en}
}
@article{DimitrovaKoppitz2022,
  author    = {Dimitrova, Ilinka and Koppitz, J{\"o}rg},
  title     = {On relative ranks of the semigroup of orientation-preserving transformations on infinite chain with restricted range},
  series = {Communications in algebra},
  volume    = {50},
  journal   = {Communications in algebra},
  number    = {5},
  publisher = {Taylor \& Francis Group},
  address   = {Philadelphia},
  issn      = {0092-7872},
  doi       = {10.1080/00927872.2021.2000998},
  pages     = {2157 -- 2168},
  year      = {2022},
  abstract  = {Let X be an infinite linearly ordered set and let Y be a nonempty subset of X. We calculate the relative rank of the semigroup OP(X,Y) of all orientation-preserving transformations on X with restricted range Y modulo the semigroup O(X,Y) of all order-preserving transformations on X with restricted range Y. For Y = X, we characterize the relative generating sets of minimal size.},
  language  = {en}
}
@article{DimitrovaFernandesKoppitz2017,
  author    = {Dimitrova, Ilinka and Fernandes, Vitor H. and Koppitz, J{\"o}rg},
  title     = {A note on generators of the endomorphism semigroup of an infinite countable chain},
  series = {Journal of Algebra and its Applications},
  volume    = {16},
  journal   = {Journal of Algebra and its Applications},
  number    = {2},
  publisher = {World Scientific},
  address   = {Singapore},
  issn      = {0219-4988},
  doi       = {10.1142/S0219498817500311},
  pages     = {9},
  year      = {2017},
  abstract  = {In this note, we consider the semigroup O(X) of all order endomorphisms of an infinite chain X and the subset J of O(X) of all transformations alpha such that vertical bar Im(alpha)vertical bar = vertical bar X vertical bar. For an infinite countable chain X, we give a necessary and sufficient condition on X for O(X) = < J > to hold. We also present a sufficient condition on X for O(X) = < J > to hold, for an arbitrary infinite chain X.},
  language  = {en}
}
@article{DimitrovaKoppitz2017,
  author    = {Dimitrova, Ilinka and Koppitz, J{\"o}rg},
  title     = {On the semigroup of all partial fence-preserving injections on a finite set},
  series = {Journal of Algebra and Its Applications},
  volume    = {16},
  journal   = {Journal of Algebra and Its Applications},
  number    = {12},
  publisher = {World Scientific},
  address   = {Singapore},
  issn      = {0219-4988},
  doi       = {10.1142/S0219498817502231},
  pages     = {14},
  year      = {2017},
  abstract  = {For n∈N , let Xn={a1,a2,…,an} be an n-element set and let F=(Xn;<f) be a fence, also called a zigzag poset. As usual, we denote by In the symmetric inverse semigroup on Xn. We say that a transformation α∈In is fence-preserving if x<fy implies that xα<fyα, for all x,y in the domain of α. In this paper, we study the semigroup PFIn of all partial fence-preserving injections of Xn and its subsemigroup IFn={α∈PFIn:α-1∈PFIn}. Clearly, IFn is an inverse semigroup and contains all regular elements of PFIn. We characterize the Green's relations for the semigroup IFn. Further, we prove that the semigroup IFn is generated by its elements with rank ≥n-2. Moreover, for n∈2N, we find the least generating set and calculate the rank of IFn.},
  language  = {en}
}
@article{FernandesKoppitzMusunthia2019,
  author    = {Fernandes, Vitor H. and Koppitz, J{\"o}rg and Musunthia, Tiwadee},
  title     = {The Rank of the Semigroup of All Order-Preserving Transformations on a Finite Fence},
  series = {Bulletin of the Malaysian Mathematical Sciences Society volume},
  volume    = {42},
  journal   = {Bulletin of the Malaysian Mathematical Sciences Society volume},
  number    = {5},
  publisher = {Malaysian mathematical sciences sciences soc},
  address   = {Pulau Punang},
  issn      = {0126-6705},
  doi       = {10.1007/s40840-017-0598-1},
  pages     = {2191 -- 2211},
  year      = {2019},
  abstract  = {A zig-zag (or fence) order is a special partial order on a (finite) set. In this paper, we consider the semigroup TFn of all order-preserving transformations on an n-element zig-zag-ordered set. We determine the rank of TFn and provide a minimal generating set for TFn. Moreover, a formula for the number of idempotents in TFn is given.},
  language  = {en}
}
@article{DeneckeKoppitzŠtrakov2006,
  author    = {Denecke, Klaus-Dieter and Koppitz, J{\"o}rg and Štrakov, Slavčo},
  title     = {Multi-hypersubstitutions and colored solid varieties},
  series = {International journal of algebra and computation},
  volume    = {16},
  journal   = {International journal of algebra and computation},
  number    = {4},
  publisher = {World Scient. Publ.},
  address   = {Singapore},
  issn      = {0218-1967},
  doi       = {10.1142/S0218196706003189},
  pages     = {797 -- 815},
  year      = {2006},
  abstract  = {Hypersubstitutions are mappings which map operation symbols to terms. Terms can be visualized by trees. Hypersubstitutions can be extended to mappings defined on sets of trees. The nodes of the trees, describing terms, are labelled by operation symbols and by colors, i.e. certain positive integers. We are interested in mappings which map differently-colored operation symbols to different terms. In this paper we extend the theory of hypersubstitutions and solid varieties to multi-hypersubstitutions and colored solid varieties. We develop the interconnections between such colored terms and multihypersubstitutions and the equational theory of Universal Algebra. The collection of all varieties of a given type forms a complete lattice which is very complex and difficult to study; multi-hypersubstitutions and colored solid varieties offer a new method to study complete sublattices of this lattice.},
  language  = {en}
}
@article{MusunthiaKoppitz2017,
  author    = {Musunthia, Tiwadee and Koppitz, J{\"o}rg},
  title     = {Maximal subsemigroups of some semigroups of order-preserving mappings on a countably infinite set},
  series = {Forum mathematicum},
  volume    = {29},
  journal   = {Forum mathematicum},
  publisher = {De Gruyter},
  address   = {Berlin},
  issn      = {0933-7741},
  doi       = {10.1515/forum-2015-0093},
  pages     = {971 -- 984},
  year      = {2017},
  abstract  = {In this paper, we study the maximal subsemigroups of several semigroups of order-preserving transformations on the natural numbers and the integers, respectively. We determine all maximal subsemigroups of the monoid of all order-preserving injections on the set of natural numbers as well as on the set of integers. Further, we give all maximal subsemigroups of the monoid of all bijections on the integers. For the monoid of all order-preserving transformations on the natural numbers, we classify also all its maximal subsemigroups, containing a particular set of transformations.},
  language  = {en}
}
@article{TinpunKoppitz2016,
  author    = {Tinpun, Kittisak and Koppitz, J{\"o}rg},
  title     = {Generating sets of infinite full transformation semigroups with restricted range},
  series = {Acta scientiarum mathematicarum},
  volume    = {82},
  journal   = {Acta scientiarum mathematicarum},
  publisher = {Institutum Bolyaianum Universitatis Szegediensis},
  address   = {Szeged},
  issn      = {0001-6969},
  doi       = {10.14232/actasm-015-502-4},
  pages     = {55 -- 63},
  year      = {2016},
  abstract  = {In the present paper, we consider minimal generating sets of infinite full transformation semigroups with restricted range modulo specific subsets. In particular, we determine relative ranks.},
  language  = {en}
}
@article{ShtrakovKoppitz2016,
  author    = {Shtrakov, Slavcho and Koppitz, J{\"o}rg},
  title     = {Stable varieties of semigroups and groupoids},
  series = {Algebra universalis},
  volume    = {75},
  journal   = {Algebra universalis},
  publisher = {Springer},
  address   = {Basel},
  issn      = {0002-5240},
  doi       = {10.1007/s00012-015-0359-7},
  pages     = {85 -- 106},
  year      = {2016},
  abstract  = {The paper deals with Sigma-composition and Sigma-essential composition of terms which lead to stable and s-stable varieties of algebras. A full description of all stable varieties of semigroups, commutative and idempotent groupoids is obtained. We use an abstract reduction system which simplifies the presentations of terms of type tau - (2) to study the variety of idempotent groupoids and s-stable varieties of groupoids. S-stable varieties are a variation of stable varieties, used to highlight replacement of subterms of a term in a deductive system instead of the usual replacement of variables by terms.},
  language  = {en}
}
@article{Koppitz2015,
  author    = {Koppitz, J{\"o}rg},
  title     = {Separation of O-n from its proper subsemigroups by a single identity},
  series = {Semigroup forum},
  volume    = {91},
  journal   = {Semigroup forum},
  number    = {1},
  publisher = {Springer},
  address   = {New York},
  issn      = {0037-1912},
  doi       = {10.1007/s00233-014-9674-0},
  pages     = {128 -- 138},
  year      = {2015},
  abstract  = {For each , we construct an identity that fails in the semigroup of all order-preserving maps on the -element chain but holds in each proper subsemigroup of O-n.},
  language  = {en}
}
@article{KoppitzMusunthia2014,
  author    = {Koppitz, J{\"o}rg and Musunthia, Tiwadee},
  title     = {Maximal subsemigroups containing a particular semigroup},
  series = {Mathematica Slovaca},
  volume    = {64},
  journal   = {Mathematica Slovaca},
  number    = {6},
  publisher = {De Gruyter},
  address   = {Warsaw},
  issn      = {0139-9918},
  doi       = {10.2478/s12175-014-0280-0},
  pages     = {1369 -- 1380},
  year      = {2014},
  abstract  = {We characterize maximal subsemigroups of the monoid T(X) of all transformations on the set X = a"center dot of natural numbers containing a given subsemigroup W of T(X) such that T(X) is finitely generated over W. This paper gives a contribution to the characterization of maximal subsemigroups on the monoid of all transformations on an infinite set.},
  language  = {en}
}
@article{DimitrovaKoppitz2011,
  author    = {Dimitrova, Ilinka and Koppitz, J{\"o}rg},
  title     = {On the maximal regular subsemigroups of ideals of order-preserving or order-reversing transformations},
  series = {Semigroup forum},
  volume    = {82},
  journal   = {Semigroup forum},
  number    = {1},
  publisher = {Springer},
  address   = {New York},
  issn      = {0037-1912},
  doi       = {10.1007/s00233-010-9272-8},
  pages     = {172 -- 180},
  year      = {2011},
  abstract  = {We characterize the maximal regular subsemigroups of the ideals of the semigroup of all order-preserving transformations as well as of the semigroup of all order-preserving or order-reversing transformations on a finite ordered set.},
  language  = {en}
}
@article{DimitrovaKoppitz2012,
  author    = {Dimitrova, Ilinka and Koppitz, J{\"o}rg},
  title     = {On the monoid of all partial order-preserving extensive transformations},
  series = {Communications in algebra},
  volume    = {40},
  journal   = {Communications in algebra},
  number    = {5},
  publisher = {Taylor \& Francis Group},
  address   = {Philadelphia},
  issn      = {0092-7872},
  doi       = {10.1080/00927872.2011.557813},
  pages     = {1821 -- 1826},
  year      = {2012},
  abstract  = {A partial transformation alpha on an n-element chain X-n is called order-preserving if x <= y implies x alpha <= y alpha for all x, y in the domain of alpha and it is called extensive if x <= x alpha for all x in the domain of alpha. The set of all partial order-preserving extensive transformations on X-n forms a semiband POEn. We determine the maximal subsemigroups as well as the maximal subsemibands of POEn.},
  language  = {en}
}
@article{DimitrovaFernandesKoppitz2012,
  author    = {Dimitrova, Ilinka and Fernandes, Vitor H. and Koppitz, J{\"o}rg},
  title     = {The maximal subsemigroups of semigroups of transformations preserving or reversing the orientation on a finite chain},
  series = {Publicationes mathematicae},
  volume    = {81},
  journal   = {Publicationes mathematicae},
  number    = {1-2},
  publisher = {Institutum Mathematicum Universitatis Debreceniensis, Debreceni Tudom{\´a}nyegyetem Matematikai Int{\´e}zete},
  address   = {Debrecen},
  issn      = {0033-3883},
  doi       = {10.5486/PMD.2012.4897},
  pages     = {11 -- 29},
  year      = {2012},
  abstract  = {The study of the semigroups OPn, of all orientation-preserving transformations on an n-element chain, and ORn, of all orientation-preserving or orientation-reversing transformations on an n-element chain, has began in [17] and [5]. In order to bring more insight into the subsemigroup structure of OPn and ORn, we characterize their maximal subsemigroups.},
  language  = {en}
}
@article{KoppitzSupaporn2013,
  author    = {Koppitz, J{\"o}rg and Supaporn, Worakrit},
  title     = {Categary equivalences of clones of operations preserving unaryoperations},
  series = {COMPTES RENDUS DE L ACADEMIE BULGARE DES SCIENCES},
  volume    = {66},
  journal   = {COMPTES RENDUS DE L ACADEMIE BULGARE DES SCIENCES},
  number    = {2},
  publisher = {Publ. House of the Bulgarian Acad. of Sciences},
  address   = {Sofia},
  issn      = {1310-1331},
  pages     = {177 -- 184},
  year      = {2013},
  abstract  = {Any clones on arbitrary set A can be written of the form Pol (A)Q for some set Q of relations on A. We consider clones of the form Pal (A)Q where Q is a set of unary relations on a finite set A. A clone Pol (A)Q is said to be a clone on a set of the smallest cardinality with respect to category equivalence if vertical bar A vertical bar <= vertical bar S vertical bar for all finite sets S and all clones C on S that category equivalent to Pol (A)Q. We characterize the clones on a set of the smallest cardinality with respect to category equivalent and show how we can find a clone on a set of the smallest cardinality that category equivalent to a given clone.},
  language  = {en}
}
@article{KarpuzCevikKoppitzetal.2013,
  author    = {Karpuz, Eylem Guzel and Cevik, Ahmet Sinan and Koppitz, J{\"o}rg and Cangul, Ismail Naci},
  title     = {Some fixed-point results on (generalized) Bruck-Reilly *-extensions of monoids},
  series = {Fixed point theory and applications},
  journal   = {Fixed point theory and applications},
  number    = {3},
  publisher = {Springer},
  address   = {Cham},
  issn      = {1687-1812},
  doi       = {10.1186/1687-1812-2013-78},
  pages     = {9},
  year      = {2013},
  abstract  = {In this paper, we determine necessary and sufficient conditions for Bruck-Reilly and generalized Bruck-Reilly *-extensions of arbitrary monoids to be regular, coregular and strongly pi-inverse. These semigroup classes have applications in various field of mathematics, such as matrix theory, discrete mathematics and p-adic analysis (especially in operator theory). In addition, while regularity and coregularity have so many applications in the meaning of boundaries (again in operator theory), inverse monoids and Bruck-Reilly extensions contain a mixture fixed-point results of algebra, topology and geometry within the purposes of this journal.},
  language  = {en}
}
@article{DimitrovaKoppitz2010,
  author    = {Dimitrova, Ilinka and Koppitz, J{\"o}rg},
  title     = {On some anti-inverse transformation semigroups},
  issn      = {1310-1331},
  year      = {2010},
  abstract  = {A semigroup S is called anti-inverse if for all a E S there is a b is an element of S such that aba = b and bab = a. Each anti-inverse semigroup is regular. In the present paper, we study anti-inverse subsemigroups within the semigroup T-n of all transformations on an n-element set (1 <= n is an element of N). In particular, we characterize all anti-inverse semigroups within the J-classes of T-n and illustrate our result by four examples.},
  language  = {en}
}
@article{Koppitz2009,
  author    = {Koppitz, J{\"o}rg},
  title     = {All Reg-solid varieties of commutative semigroups},
  issn      = {0037-1912},
  doi       = {10.1007/s00233-008-9124-y},
  year      = {2009},
  abstract  = {We determine all regular solid varieties of commutative semigroups. Each of them is contained in the Reg- hyperequational class V (RC) defined by the associative law and the commutative law, and every subvariety of V (RC) is regular solid. In the present paper, the subvariety lattice of V (RC) will be characterized.},
  language  = {en}
}
@phdthesis{Koppitz1993,
  author    = {Koppitz, J{\"o}rg},
  title     = {{\"U}ber Halbgruppen mit vereinigungshalbdistributivem Unterhalbgruppenverband},
  pages     = {60 S.},
  year      = {1993},
  language  = {de}
}
@article{DeneckeKoppitz1994,
  author    = {Denecke, Klaus-Dieter and Koppitz, J{\"o}rg},
  title     = {Hyperassociative semigroups},
  year      = {1994},
  language  = {en}
}
@article{DeneckeKoppitz1995,
  author    = {Denecke, Klaus-Dieter and Koppitz, J{\"o}rg},
  title     = {Pre-solid varieties of commutative semigroups},
  year      = {1995},
  language  = {en}
}
@article{DeneckeKoppitz1995,
  author    = {Denecke, Klaus-Dieter and Koppitz, J{\"o}rg},
  title     = {Pre-solid varieties of semigroups},
  year      = {1995},
  language  = {en}
}
@article{DeneckeKoppitz1995,
  author    = {Denecke, Klaus-Dieter and Koppitz, J{\"o}rg},
  title     = {M-solid varieties of semigroups},
  year      = {1995},
  language  = {en}
}
@article{Koppitz1997,
  author    = {Koppitz, J{\"o}rg},
  title     = {M-solid subvarieties of some varieties of commutative semigroups},
  year      = {1997},
  language  = {en}
}
@article{WismathKoppitzDenecke1997,
  author    = {Wismath, Shelly and Koppitz, J{\"o}rg and Denecke, Klaus-Dieter},
  title     = {Maps between M-solid varieties of emigroups},
  year      = {1997},
  language  = {en}
}
@article{DeneckeKoppitzMarszalek1998,
  author    = {Denecke, Klaus-Dieter and Koppitz, J{\"o}rg and Marszalek, R.},
  title     = {Derived varieties and derived equational theories},
  year      = {1998},
  language  = {en}
}
@article{DeneckeKoppitz1998,
  author    = {Denecke, Klaus-Dieter and Koppitz, J{\"o}rg},
  title     = {M-solid monoids of hypersubstitutions of type 2},
  year      = {1998},
  language  = {en}
}
@article{DeneckeKoppitz1998,
  author    = {Denecke, Klaus-Dieter and Koppitz, J{\"o}rg},
  title     = {Finite monoids of hypersubstitutions of type € = (2)},
  year      = {1998},
  language  = {en}
}
@article{DeneckeKoppitz1999,
  author    = {Denecke, Klaus-Dieter and Koppitz, J{\"o}rg},
  title     = {Normal forms of hypersubstitutions},
  year      = {1999},
  language  = {en}
}
@article{DeneckeKoppitz1999,
  author    = {Denecke, Klaus-Dieter and Koppitz, J{\"o}rg},
  title     = {A characterization of M-solid varieties of semigroups},
  year      = {1999},
  language  = {en}
}
@article{Koppitz2000,
  author    = {Koppitz, J{\"o}rg},
  title     = {All 2-solid varieties of semigroups},
  year      = {2000},
  language  = {en}
}
@article{DeneckeKoppitz2000,
  author    = {Denecke, Klaus-Dieter and Koppitz, J{\"o}rg},
  title     = {Essential variables in weak hypersubstitutions},
  isbn      = {3-8265- 7983-6},
  year      = {2000},
  language  = {en}
}
@article{DeneckeKoppitzWismath2001,
  author    = {Denecke, Klaus-Dieter and Koppitz, J{\"o}rg and Wismath, Shelly},
  title     = {The semantical hyperunification problem},
  year      = {2001},
  language  = {en}
}
@article{ArwornDeneckeKoppitz2001,
  author    = {Arworn, Srichan and Denecke, Klaus-Dieter and Koppitz, J{\"o}rg},
  title     = {Strongly luid and weakly unsolid varieties},
  issn      = {1346-0862},
  year      = {2001},
  language  = {en}
}
@article{DeneckeKoppitz2001,
  author    = {Denecke, Klaus-Dieter and Koppitz, J{\"o}rg},
  title     = {Essential variables in hypersubstitution},
  year      = {2001},
  language  = {en}
}
@book{DeneckeKoppitzShtraklov2001,
  author    = {Denecke, Klaus-Dieter and Koppitz, J{\"o}rg and Shtraklov, Slavcho},
  title     = {The Depth of a Hypersubstitution},
  year      = {2001},
  language  = {en}
}
@article{DeneckeKoppitz2001,
  author    = {Denecke, Klaus-Dieter and Koppitz, J{\"o}rg},
  title     = {Fluid, unsolid and completely unsolid varieties},
  year      = {2001},
  language  = {en}
}
@phdthesis{Koppitz2001,
  author    = {Koppitz, J{\"o}rg},
  title     = {M-solide Variet{\"a}ten von Halbgruppen},
  pages     = {183 S.},
  year      = {2001},
  language  = {de}
}
@article{DeneckeKoppitzWismath2002,
  author    = {Denecke, Klaus-Dieter and Koppitz, J{\"o}rg and Wismath, Shelly},
  title     = {Solid Varietie of Arbitrary Type},
  year      = {2002},
  language  = {en}
}
@article{DeneckeKoppitzNiwczyk2002,
  author    = {Denecke, Klaus-Dieter and Koppitz, J{\"o}rg and Niwczyk, St.},
  title     = {Equational theories generated by generalized hypersubstitutions of type (n)},
  year      = {2002},
  language  = {en}
}