In this paper we study the semigroup $\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N})$ of partial cofinal monotone bijective transformations of the set of positive integers $\mathbb{N}$. We show that the semigroup $\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N})$ has algebraic properties similar to the bicyclic semigroup: it is bisimple and all of its non-trivial group homomorphisms are either isomorphisms or group homomorphisms. We also prove that every locally compact topology $\tau$ on $\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N})$ such that $(\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N}),\tau)$ is a topological inverse semigroup, is discrete. Finally, we describe the closure of $(\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N}),\tau)$ in a topological semigroup.
Deep Dive into Topological monoids of monotone injective partial selfmaps of $mathbb{N}$ with cofinite domain and image.
In this paper we study the semigroup $\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N})$ of partial cofinal monotone bijective transformations of the set of positive integers $\mathbb{N}$. We show that the semigroup $\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N})$ has algebraic properties similar to the bicyclic semigroup: it is bisimple and all of its non-trivial group homomorphisms are either isomorphisms or group homomorphisms. We also prove that every locally compact topology $\tau$ on $\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N})$ such that $(\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N}),\tau)$ is a topological inverse semigroup, is discrete. Finally, we describe the closure of $(\mathscr{I}_{\infty}^{\nearrow}(\mathbb{N}),\tau)$ in a topological semigroup.
Our purpose is to study the semigroup I ึ โ (N) of partial cofinal monotone bijective transformations of the set of positive integers N. We shall show that the semigroup I ึ โ (N) has algebraic properties similar to the bicyclic semigroup: it is bisimple and all of its nontrivial group homomorphisms are either isomorphisms or group homomorphisms. We shall also prove that every locally compact topology ฯ on I ึ โ (N) such that (I ึ โ (N), ฯ ) is a topological inverse semigroup is discrete and we shall describe the closure of (I ึ โ (N), ฯ ) in a topological semigroup. In this paper all spaces will be assumed to be Hausdorff. Furthermore we shall follow the terminology of [5,6,8]. We shall denote the first infinite cardinal by ฯ and the cardinality of the set A by |A|. If Y is a subspace of a topological space X and A โ Y , then we shall denote the topological closure and the interior of A in Y by cl Y (A) and Int Y (A), respectively.
An algebraic semigroup S is called inverse if for any element x โ S there exists a unique x -1 โ S such that xx -1 x = x and x -1 xx -1 = x -1 . The element x -1 is called the inverse of x โ S. If S is an inverse semigroup, then the function inv : S โ S which assigns to every element x of S its inverse element x -1 is called an inversion.
If S is a semigroup, then we shall denote the band (i. e. the subset of idempotents) of S by E(S). If the band E(S) is a nonempty subset of S, then the semigroup operation on S determines the partial order on E(S): e f if and only if ef = f e = e. This order is called natural.
A semilattice is a commutative semigroup of idempotents. A semilattice E is called linearly ordered or chain if the semilattice operation admits a linear natural order on E. A maximal chain of a semilattice E is a chain which is properly contained in no other chain of E. The Axiom of Choice implies the existence of maximal chains in any partially ordered set. According to [11,Definition II.5.12] a chain L is called an ฯ-chain if L is isomorphic to {0, -1, -2, -3, . . .} with the usual order . Let E be a semilattice and e โ E. We denote โe = {f โ E | f e} and โe = {f โ E | e f }.
A topological (inverse) semigroup is a topological space together with a continuous multiplication (and an inversion, respectively). Let I ฮป denote the set of all partial one-to-one transformations of a set X of cardinality ฮป together with the following semigroup operation:
The semigroup I ฮป is called the symmetric inverse semigroup over the set X (see [6]). The symmetric inverse semigroup was introduced by Wagner [12] and it plays a major role in the theory of semigroups.
Let N be the set of all positive integers. We shall denote the semigroup of monotone, non-decreasing, injective partial transformations of N such that the sets N \ dom ฯ and N \ rank ฯ are finite for all ฯ โ I ึ โ (N) by (ii) n 1 < n 2 < n 3 < n 4 < . . . and m 1 < m 2 < m 3 < m 4 < . . . . We observe that an element ฮฑ of the semigroup I ฯ is an element of the semigroup I ึ โ (N) if and only if it satisfies the conditions (i) and (ii).
The bicyclic semigroup C (p, q) is the semigroup with the identity 1, generated by elements p and q, subject only to the condition pq = 1. The bicyclic semigroup is bisimple and every one of its congruences is either trivial or a group congruence. Moreover, every non-annihilating homomorphism h of the bicyclic semigroup is either an isomorphism or the image of C (p, q) under h is a cyclic group (see [6,Corollary 1.32]). The bicyclic semigroup plays an important role in algebraic theory of semigroups and in the theory of topological semigroups.
For example, the well-known result of Andersen [1] states that a (0-)simple semigroup is completely (0-)simple if and only if it does not contain the bicyclic semigroup. The bicyclic semigroup admits only the discrete topology and a topological semigroup S can contain C (p, q) only as an open subset [7]. Neither stable nor ฮ-compact topological semigroups can contain a copy of the bicyclic semigroup [2,10]. Also, the bicyclic semigroup does not embed into a countably compact topological inverse semigroup [9].
Moreover, the conditions were given in [3] and [4] when a countable compact or pseudocompact topological semigroup does not contain the bicyclic semigroup. However, Banakh, Dimitrova and Gutik constructed, using set-theoretic assumptions (Continuum Hypothesis or Martin Axiom), an example of a Tychonoff countably compact topological semigroup which contains the bicyclic semigroup [4].
We remark that the bicyclic semigroup is isomorphic to the semigroup C N (ฮฑ, ฮฒ) which is generated by partial transformations ฮฑ and ฮฒ of the set of positive integers N, defined as follows:
Therefore the semigroup I ึ โ (N) contains an isomorphic copy of the bicyclic semigroup C (p, q).
- Algebraic properties of the semigroup I ึ โ (N) Proposition 2.1. The following properties hold:
Statement (ii) follows from definitions of relations R and L and Theorem 1.17 o
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