Free Semilattice Research Articles

On the monoid of cofinite partial isometries of $\qq{N}^n$ with the usual metric

In this paper we study the structure of the monoid Iℕn ∞ of cofinite partial isometries of the n-th power of the set of positive integers ℕ with the usual metric for a positive integer n > 2. We describe the group of units and the subset of idempotents of the semigroup Iℕn ∞, the natural partial order and Green's relations on Iℕn ∞. In particular we show that the quotient semigroup Iℕn ∞/Cmg, where Cmg is the minimum group congruence on Iℕn ∞, is isomorphic to the symmetric group Sn and D = J in Iℕn ∞. Also, we prove that for any integer n ≥2 the semigroup Iℕn ∞ is isomorphic to the semidirect product Sn ×h(P∞(Nn); U) of the free semilattice with the unit (P∞(Nn); U) by the symmetric group Sn.

On the zero-divisor graphs of finite free semilattices

Let $SL_{X}$ be the free semilattice on a finite nonempty set $X$. There exists an undirected graph $\Gamma(SL_{X})$ associated with $SL_{X}$ whose vertices are the proper subsets of $X$, except the empty set, and two distinct vertices $A$ and $B$ of $\Gamma(SL_{X})$ are adjacent if and only if $A\cup B=X$. In this paper, the diameter, radius, girth, degree of any vertex, domination number, independence number, clique number, chromatic number, and chromatic index of $\Gamma(SL_{X})$ have been established. Moreover, we have determined when $\Gamma(SL_{X})$ is a perfect graph and when the core of $\Gamma(SL_{X})$ is a Hamiltonian graph.

Elements of Algebraic Geometry Over a Free Semilattice

It is proved that every consistent system of equations over a free semilattice of arbitrary rank is equivalent to its finite subsystem. Furthermore, irreducible algebraic sets are studied, and we look at the consistency problem for systems of equations over free semilattices.

The Lattice of Subvarieties of Semilattice Ordered Algebras

This paper is devoted to the semilattice ordered \(\mathcal{V}\)-algebras of the form (A, Ω, + ), where + is a join-semilattice operation and (A, Ω) is an algebra from some given variety \(\mathcal{V}\). We characterize the free semilattice ordered algebras using the concept of extended power algebras. Next we apply the result to describe the lattice of subvarieties of the variety of semilattice ordered \(\mathcal{V}\)-algebras in relation to the lattice of subvarieties of the variety \(\mathcal{V}\).

Rank properties of certain semigroups

In this paper we consider certain ranks of some semigroups. These ranks are r1(S),r2(S),r3(S),r4(S) and r5(S) as defined below. We have r1≤r2≤r3≤r4≤r5. The semigroups are CLn,CLm×CLn,Zn and SLn. Here CLn is a chain with n elements, Zn is the zero semigroup on n elements and SLn is the free semilattice generated by n elements and having 2n−1 elements. We find many of the ranks for these classes of semigroups.

Inside Every Model of Abstract Stone Duality Lies an Arithmetic Universe

The first paper published on Abstract Stone Duality showed that the overt discrete objects (those admitting ∃ and = internally) form a pretopos, i.e. a category with finite limits, stable disjoint coproducts and stable effective quotients of equivalence relations. Using an N-indexed least fixed point axiom, here we show that this full subcategory is an arithmetic universe, having a free semilattice (“collection of Kuratowski-finite subsets”) and a free monoid (“collection of lists”) on any overt discrete object. Each finite subset is represented by its pair (□, ◊) of modal operators, although a tight correspondence with these depends on a stronger Scott-continuity axiom. Topologically, such subsets are both compact and open and also involve proper open maps. In applications of ASD this can eliminate lists in favour of a continuation-passing interpretation.

Minimal Presentations and Embedding into Inefficient Semigroups

In this paper, we show that CLn, the chain with n elements, is efficient and that the direct product CLm× CLnis inefficient. Moreover, we embed any finitely presented semigroup S into an inefficient semigroup, namely, the semigroup S ⋃ SLn, where SLnis the free semilattice of rank n.

Minimal Presentations and Efficiency of Semigroups

is said to be efficient if it can be defined by a presentation (A | R) with |R| -|A|=rank(H 2(S)). In this paper we demonstrate certain infinite classes of both efficient and inefficient semigroups. Thus, finite abelian groups, dihedral groups D 2 n with n even, and finite rectangular bands are efficient semigroups. By way of contrast we show that finite zero semigroups and free semilattices are never efficient. These results are compared with some well-known results on the efficiency of groups.

On subsemigroup lattices without non-trivial identities

A latticeL is called discriminating if for any free latticeF and for any finite number of elementsu1,u2, ...,un∈F, there exists a homomorphismf:F→L such thatf(ui)≠f(uj) wheneverui ≠ uj (1≤i, j≤n). In this paper it is proved that the subsemigroup lattice SubS of a commutative semigroupS does not satisfy a non-trivial identity if and only if SubS is discriminating. In particular, in this case every finite projective lattice can be embedded into SubS. It should be noted that the most important examples of semigroups whose subsemigroup lattices satisfy no non-trivial identity and therefore have the discriminating property are the following: the infinite cyclic semigroup, the free semilattice of countable rank, any commutative nilsemigroup which is not nilpotent and so on.

The Bifree Locally Inverse Semigroup on a Set

A class of regular semigroups closed under taking direct products, regular subsemigroups and homomorphic images is an e(xistence) variety of regular semigroups. For an e-variety V of locally inverse or E-solid regular semigroups, the bifree object BFV(X) on a set X is the natural concept of a "free object" in V. Its existence has been proved by Y. T. Yeh. In this paper, the bifree locally inverse semigroup BFLJ(X) is described as a homomorphic image of the absolutely free algebra of type [2, 2] generated by X and the set of formal inverses X′, and equivalently as subsemigroup of a semidirect product of a suitable free semilattice by the bifree completely simple semigroup on X. This latter realization is used to show that BFLJ(X) is combinatorial, completely semisimple and satisfies several finiteness conditions. Furthermore, the approach of biidentities is used to formulate a Birkhoff-type theorem for e-varieties of locally inverse semigroups and to establish a one-one correspondence between locally inverse e-varieties and fully invariant congruences on BFLJ(X) for countably infinite X. As an application, it is shown that in each e-variety of locally inverse semigroups all free products exist.

Free generators in relatively free completely regular semigroups

SynopsisA subset Y of a free completely regular semigroup FCRx freely generates a free completely regular subsemigroup if and only if (i) each -class of FCRx contains at most one element of Y, (ii) {Dy;y ∊ Y} freely generates a free subsemilattice of the free semilattice FCRx/), and (iii) Y consists of non-idempotents. A similar description applies in free objects of some subvarieties of the variety of all completely regular semigroups.