Semigroups of linear tree languages

2018 ◽  
Vol 11 (06) ◽  
pp. 1850091
Author(s):  
Pongsakorn Kitpratyakul ◽  
Bundit Pibaljommee
Keyword(s):  
Green’S Relations ◽  
Tree Language ◽  
Type Formula ◽  
Product Operation ◽  
Green's Relations ◽  
Tree Languages ◽  
The Universe ◽  
Linear Product

A linear tree language of type [Formula: see text] is a set of linear terms, terms in which each variable occurs at most once, of that type. We investigate a semigroup consisting of the collection of all linear tree languages such that products of any element in the collection are nonempty and the operation of the corresponding linear product especially idempotent elements, Green’s relations [Formula: see text], [Formula: see text], and [Formula: see text], and some of its subsemigroups. We discover that this semigroup is neither factorizable nor locally factorizable. We also study the linear product iteration and show that any iteration is idempotent in this semigroup. Moreover, we study a semigroup with the complement of the universe set of the above semigroup together with the same linear product operation.

A generalized superposition of linear tree languages and products of linear tree languages

2018 ◽  
Vol 11 (04) ◽  
pp. 1850048
Author(s):  
Pongsakorn Kitpratyakul ◽  
Bundit Pibaljommee
Keyword(s):  
Green’S Relations ◽  
Linear Superposition ◽  
Tree Language ◽  
Regular Elements ◽  
Type Formula ◽  
Green's Relations ◽  
Tree Languages

A linear tree language of type [Formula: see text] is a set of linear terms, terms containing no multiple occurrences of the same variable, of that type. Instead of the usual generalized superposition of tree languages, we define the generalized linear superposition to deal with linear tree languages and study its properties. Using this superposition, we define the product of linear tree languages. This product is not associative on the collection of all linear tree languages, but it is associative on some subsets of this collection whose products of any element in the subsets are nonempty. We classify such subsets and study properties of the obtained semigroup especially idempotent elements, regular elements, and Green’s relations [Formula: see text] and [Formula: see text].

SEMIGROUPS OF TREE LANGUAGES

2008 ◽  
Vol 01 (04) ◽  
pp. 489-507 ◽  
Author(s):  
K. Denecke ◽  
N. Sarasit
Keyword(s):  
Left Zero ◽  
Green’S Relations ◽  
Regular Elements ◽  
Green's Relations ◽  
Tree Languages

Sets of terms of type τ are called tree languages (see [6]). There are several possibilities to define superposition operations on sets of tree languages. On the basis of such superposition operations we define binary associative operations on tree languages and investigate the properties of the arising semigroups. We characterize idempotent and regular elements and Green's relations [Formula: see text] and [Formula: see text]. Moreover, we determine constant, left-zero and right-zero subsemigroups and rectangular bands.

Semigroup properties of linear terms

2017 ◽  
Vol 10 (01) ◽  
pp. 1750051 ◽  
Author(s):  
Laddawan Lohapan ◽  
Prakit Jampachon
Keyword(s):  
Galois Theory ◽  
Green’S Relations ◽  
Regular Elements ◽  
Type Formula ◽  
Green's Relations ◽  
Partial Algebras

The concept of linear terms of a given type [Formula: see text] (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 [Formula: see text] and [Formula: see text] on the set [Formula: see text] of all linear terms of type [Formula: see text] Then we characterize regular elements and Green’s relations in these partial algebras.

Semigroups of an inductive composition of terms

2021 ◽  
pp. 2250038
Author(s):  
Pongsakorn Kitpratyakul ◽  
Bundit Pibaljommee
Keyword(s):  
Regularity Condition ◽  
Binary Operation ◽  
Green’S Relations ◽  
Algebraic Structures ◽  
Type Formula ◽  
Green's Relations ◽  
Composition Of Terms

The set of all [Formula: see text]-ary terms of type [Formula: see text] together with a binary operation derived from a superposition [Formula: see text] forms various forms of semigroups. One may generalize such binary operation by deriving it from an inductive composition of terms and call it an inductive product. However, this operation is not associative on the same base set but it becomes associative when all elements of subterms of a fixed term used in an inductive product except itself are excluded from the base set. Hence, a semigroup is formed. In this paper, we mainly focus on the algebraic structures of this semigroup such as idempotent elements, elements associating with each type of regularity condition, and Green’s relations. The formulae of complexity of inducted terms are also under investigation.

scholarly journals ALGEBRAIC CHARACTERIZATION OF LOGICALLY DEFINED TREE LANGUAGES

2010 ◽  
Vol 20 (02) ◽  
pp. 195-239 ◽  
Author(s):  
ZOLTÁN ÉSIK ◽  
PASCAL WEIL
Keyword(s):  
Algebraic Structure ◽  
Algebraic Characterization ◽  
Tree Language ◽  
Regular Tree ◽  
First Order ◽  
Product Operation ◽  
Finite Word ◽  
Tree Languages ◽  
Block Product

We give an algebraic characterization of the tree languages that are defined by logical formulas using certain Lindström quantifiers. An important instance of our result concerns first-order definable tree languages. Our characterization relies on the usage of preclones, an algebraic structure introduced by the authors in a previous paper, and of the block product operation on preclones. Our results generalize analogous results on finite word languages, but it must be noted that, as they stand, they do not yield an algorithm to decide whether a given regular tree language is first-order definable.

scholarly journals Certain characterizations of varieties of bands

1988 ◽  
Vol 31 (2) ◽  
pp. 301-319 ◽  
Author(s):  
J. A. Gerhard ◽  
Mario Petrich
Keyword(s):  
Rectangular Band ◽  
Simple System ◽  
Left Zero ◽  
Green’S Relations ◽  
Transformation Semigroups ◽  
Lattice Of Varieties ◽  
Green's Relations ◽  
The World ◽  
New System

The lattice of varieties of bands was constructed in [1] by providing a simple system of invariants yielding a solution of the world problem for varieties of bands including a new system of inequivalent identities for these varieties. References [3] and [5] contain characterizations of varieties of bands determined by identities with up to three variables in terms of Green's relations and the functions figuring in a construction of a general band. In this construction, the band is expressed as a semilattice of rectangular bands and the multiplication is written in terms of functions among these rectangular band components and transformation semigroups on the corresponding left zero and right zero direct factors.

PROPERTIES OF QUASI-RELABELING TREE BIMORPHISMS

2010 ◽  
Vol 21 (03) ◽  
pp. 257-276 ◽  
Author(s):  
ANDREAS MALETTI ◽  
CĂTĂLIN IONUŢ TÎRNĂUCĂ
Keyword(s):  
Cartesian Product ◽  
Canonical Representation ◽  
Finite Union ◽  
Top Down ◽  
Tree Language ◽  
Regular Tree ◽  
Fundamental Properties ◽  
Tree Transducers ◽  
Tree Languages ◽  
Context Free

The fundamental properties of the class QUASI of quasi-relabeling relations are investigated. A quasi-relabeling relation is a tree relation that is defined by a tree bimorphism (φ, L, ψ), where φ and ψ are quasi-relabeling tree homomorphisms and L is a regular tree language. Such relations admit a canonical representation, which immediately also yields that QUASI is closed under finite union. However, QUASI is not closed under intersection and complement. In addition, many standard relations on trees (e.g., branches, subtrees, v-product, v-quotient, and f-top-catenation) are not quasi-relabeling relations. If quasi-relabeling relations are considered as string relations (by taking the yields of the trees), then every Cartesian product of two context-free string languages is a quasi-relabeling relation. Finally, the connections between quasi-relabeling relations, alphabetic relations, and classes of tree relations defined by several types of top-down tree transducers are presented. These connections yield that quasi-relabeling relations preserve the regular and algebraic tree languages.

scholarly journals Green's relations for regular elements of sandwich semigroups, II; semigroups of continuous functions

1978 ◽  
Vol 25 (1) ◽  
pp. 45-65 ◽  
Author(s):  
K. D. Magill ◽  
S. Subbiah
Keyword(s):  
Continuous Functions ◽  
Maximal Subgroups ◽  
Green’S Relations ◽  
Boolean Ring ◽  
Regular Elements ◽  
Ring Homomorphisms ◽  
Special Cases ◽  
Green's Relations ◽  
Sandwich Semigroup ◽  
Sandwich Semigroups

AbstractA sandwich semigroup of continuous functions consists of continuous functions with domains all in some space X and ranges all in some space Y with multiplication defined by fg = foαog where α is a fixed continuous function from a subspace of Y into X. These semigroups include, as special cases, a number of semigroups previously studied by various people. In this paper, we characterize the regular elements of such semigroups and we completely determine Green's relations for the regular elements. We also determine the maximal subgroups and, finally, we apply some of these results to semigroups of Boolean ring homomorphisms.

scholarly journals Green’s Relations on a Semigroup of Transformations with Restricted Range that Preserves an Equivalence Relation and a Cross-Section

Mathematics ◽  
2018 ◽  
Vol 6 (8) ◽  
pp. 134
Author(s):  
Chollawat Pookpienlert ◽  
Preeyanuch Honyam ◽  
Jintana Sanwong
Keyword(s):  
Cross Section ◽  
Equivalence Relation ◽  
Nonempty Subset ◽  
Identity Relation ◽  
Green’S Relations ◽  
Restricted Range ◽  
Green's Relations ◽  
Semigroup Of Transformations

Let T(X,Y) be the semigroup consisting of all total transformations from X into a fixed nonempty subset Y of X. For an equivalence relation ρ on X, let ρ^ be the restriction of ρ on Y, R a cross-section of Y/ρ^ and define T(X,Y,ρ,R) to be the set of all total transformations α from X into Y such that α preserves both ρ (if (a,b)∈ρ, then (aα,bα)∈ρ) and R (if r∈R, then rα∈R). T(X,Y,ρ,R) is then a subsemigroup of T(X,Y). In this paper, we give descriptions of Green’s relations on T(X,Y,ρ,R), and these results extend the results on T(X,Y) and T(X,ρ,R) when taking ρ to be the identity relation and Y=X, respectively.

Sign in / Sign up

Export Citation Format

Share Document