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We investigate ternary products of groupoids and prove that there is a one-to-one correspondence between the collection of right modular groupoids with a left identity element l and laterally commutative, l-bi-unital semiheaps. This... more
Let be a commutative (not necessary unital) inverse semigroup with the set of idempotents then is a commutative Banach -module with canonical actions. Recently, it is shown that the triangular Banach algebra is -weakly -module... more
Let X_n={1,2,3,..,n} and Let α:Domα⊆X_n⟶Im⊆X_n be a partial one-to-one transformation on. The elements of partial one to one transformation semigroup were constructed and a Triad subsemigroup was identified using the intersections of... more
Let X_n={1,2,3,..,n} and Let α:Domα⊆X_n⟶Im⊆X_n be a partial one-to-one transformation on . The elements of partial one to one transformation semigroup were constructed and a Triad subsemigroup was identified using the intersections of... more
For n ∈ N let Xn = {1 < 2 < · · · < n} be a finite n-element chain and let Tn denote the full transformation semigroup, i.e. the semigroup of all mappings α : Xn → Xn under composition. We say that a transformation α ∈ Tn is... more
Let T n be the full symmetric semigroup on X n = {1, 2,. .. , n} and let OCT n and ORCT n be its subsemigroups of order-preserving and order-preserving or order-reversing full contraction mappings of X n , respectively. In this paper we... more
Let $[n]=\{1,2,\ldots,n\}$ be a finite chain. Let $\mathcal{P}_{n}$ and $\mathcal{T}_{n}$ be Semigroups of partial and full transformations on $[n]$ respectively. Let $\mathcal{CP}_{n}=\{\alpha\in \mathcal{P}_{n}:... more
Let [n] = {1, 2,. .. , n} be a finite chain and let CT n be the semigroup of full contractions on [n]. Denote ORCT n and OCT n to be the subsemigroup of order preserving or reversing and the subsemigroup of order preserving full... more
Let PO n be the semigroup of all order-preserving partial transformations of a finite chain. It is shown that there exist bijections between the set of certain lattice paths in the Cartesian plane that start at (0, 0), end at (n − 1, n −... more
Let Sing, be the semigroup of singular self-maps of X, = {l, ,n}, let R, = {a E Sing,,: (VJJ E Im a) 1 ya-' 1 2 1 Im al} and let E(R,) be the set of idempotents of R,. Then it is shown that R, = (E(R,))'. Moreover, expressions for the... more
We study inverse semigroup amalgams [S 1 , S 2 ; U ], where S 1 and S 2 are finitely presented inverse semigroups with decidable word problem and U is an inverse semigroup with decidable membership problem in S 1 and S 2. We use a... more
Let X_n={1,2,3,..,n} and Let α:Domα⊆X_n⟶Im⊆X_n be a partial one-to-one transformation on. The elements of partial one to one transformation semigroup were constructed and a Triad subsemigroup was identified using the intersections of... more
Let G be a permutation group on a set Q. with no fixed point in Q. If for each subset P of Q the size |r g-f| is bounded, for g € G, we define the movement of g asthemax|r*-F| over all subsets f of Q. In particular, if all non-identity... more
Let G be a permutation group on a set Ω with no fixed point in Ω. If for each subset Г of Ω the size |Гg - Г| is bounded, for g ∈ G, we define the movement of g as the max|Гg − Г| over all subsets Г of Ω. In particular, if all... more
The commuting graph of a finite non-commutative semigroup S, denoted G(S), is a simple graph whose vertices are the non-central elements of S and two distinct vertices x, y are adjacent if xy = yx. Let I(X) be the symmetric inverse... more
In a group G, elements a and b are conjugate if there exists g ∈ G such that g −1 ag = b. This conjugacy relation, which plays an important role in group theory, can be extended in a natural way to inverse semigroups: for elements a and b... more
A magma is an algebra with a binary operation •, and a Boolean magma is a Boolean algebra with an additional binary operation • that distributes over all finite Boolean joins. We prove that all square-increasing (x ≤ x 2) Boolean magmas... more
For two finite monoids S and T , we prove that the second integral homology of the Schützenberger product S3T is equal to H2(S3T) = H2(S) × H2(T) × (H1(S) ⊗ Z H1(T)) as the second integral homology of the direct product of two monoids.... more
It is well known that in every inverse semigroup the binary operation and the unary operation of inversion satisfy the following three identities: x = (xx ′)x (xx ′)(y ′ y) = (y ′ y)(xx ′) (xy)z = x(yz ′′). The goal of this note is to... more
When a semigroup has a unary operation, it is possible to define two binary operations, namely, left and right division. In addition it is well known that groups can be defined in terms of those two divisions. The aim of this paper is to... more
A groupoid S that satisfies the left invertive law, ab•c = cb•a is called an AG-groupoid. We extend this concept to introduce a Stein AG-groupoid. We prove the existence of this type of AG-groupoid by providing some non-associative... more
We introduce a general technique for finding sets of axioms for a given class of semigroups. To illustrate the technique, we provide new sets of defining axioms for groups of exponent n, bands, and semilattices.
We study inverse semigroup amalgams [S 1 , S 2 ; U ], where S 1 and S 2 are finitely presented inverse semigroups with decidable word problem and U is an inverse semigroup with decidable membership problem in S 1 and S 2. We use a... more
We generalize the regularity concept for semigroups in two ways simultaneously: to higher regularity and to higher arity. We show that the one-relational and multi-relational formulations of higher regularity do not coincide, and each... more
There exists a class of widespread languages that use stack machines for interpretation of programs, the so called stack based languages (Java virtual machine language, Forth, Postscript, several intermediate program representation... more
Constellations were recently introduced by the authors as onesided analogues of categories: a constellation is equipped with a partial multiplication for which 'domains' are defined but, in general, 'ranges' are not. Left restriction... more
We introduce the notions of a generalised category and of an inductive generalised category over a band. Our purpose is to describe a class of semigroups which we name weakly B-orthodox. In doing so we produce a new approach to... more
The Ehresmann-Schein-Nambooripad (ESN) Theorem, stating that the category of inverse semigroups and morphisms is isomorphic to the category of inductive groupoids and inductive functors, is a powerful tool in the study of inverse... more
Inductive constellations are one-sided analogues of inductive categories which correspond to left restriction semigroups in a manner analogous to the correspondence between inverse semigroups and inductive groupoids. In this paper, we... more
The Ehresmann-Schein-Nambooripad (ESN) Theorem, stating that the category of inverse semigroups and morphisms is isomorphic to the category of inductive groupoids and inductive functors, is a powerful tool in the study of inverse... more
The Ehresmann-Schein-Nambooripad (ESN) Theorem asserts an equivalence between the category of inverse semigroups and the category of inductive groupoids. In this paper, we consider the category of inverse categories and functors -- a... more
General properties of ternary semigroups and groups are considered. The bi-element representation theory in which every representation matrix corresponds to a pair of elements is built, connection with the standard theory is considered... more
We generalize the regularity concept for semigroups in two ways simultaneously: to higher regularity and to higher arity. We show that the one-relational and multi-relational formulations of higher regularity do not coincide, and each... more
We prove that four different notions of Morita equivalence for inverse semigroups motivated by, respectively, C *-algebra theory, topos theory, semigroup theory and the theory of ordered groupoids are equivalent. We also show that the... more
We investigate the relationship which exists between certain classes of ordered small categories, introduced by Charles Ehresmann in the course of his work on local structures, and the class of U-semiabundant semigroups, first studied by... more
It has recently been shown how to construct an inverse semigroup from any tiling: a construction having applications in K-theoretical gap-labelling. In this paper, we provide the categorical basis for this construction in terms of an... more
We select a new type of elements of n-ary n V 3 semigroups with an idempotent and investigate properties of ideals connected with these elements.
This paper establishes methods that quantify the structure of statistical interactions within a given data set using the characterization of information theory in cohomology by finite methods, and provides their expression in terms of... more
Constellations were recently introduced by the authors as onesided analogues of categories: a constellation is equipped with a partial multiplication for which 'domains' are defined but, in general, 'ranges' are not. Left restriction... more
Inverse semigroups form a variety of unary semigroups, that is, semigroups equipped with an additional unary operation, in this case a → a −1. The theory of inverse semigroups is perhaps the best developed within semigroup theory, and... more
A subsemigroup S of a semigroup Q is a left order in Q and Q is a semigroup of left quotients of S if every element of Q can be expressed as a ♯ b where a, b ∈ S and if, in addition, every element of S that is square cancellable lies in a... more
In this note we generalize the regularity concept for semigroups in two ways simul- taneously: to higher regularity and to higher arity. We show that the one-relational and multi-relational formulations of higher regularity do not... more
The reader is guided through a detailed proof of the Ehresmann-Schein-Nambooripad Theorem from the point of view of symmetry, following Lawson's proof in Inverse Semigroups: The Theory of Partial Symmetries. Sierpiński's Sieve is taken as... more
We propose to extend "invertibility" to "regularity" for categories in general abstract algebraic manner. Higher regularity conditions and "semicommutative" diagrams are introduced. Distinction between commutative and "semicommutative"... more
We consider a nonstandard odd reduction of supermatrices (as compared with the standard even one) which arises in connection with possible extension of manifold structure group reductions. The study was initiated by consideration of the... more
Inductive constellations are one-sided analogues of inductive categories which correspond to left restriction semigroups in a manner analogous to the correspondence between inverse semigroups and inductive groupoids. In this paper, we... more
We associate with every anti-ordered set ((X, =, =), α) with α ∩ α −1 = ∅ a partial groupoid ((X, =, =), ·) in such a way that (x, y) ∈ α ⇐⇒ x·y = y and (x, y) α ⇐⇒ x · y = x for two elements x, y ∈ X such that x = y.