Amalgamation for inverse and generalized inverse semigroups

It is well-known that on an inverse semigroup S the relation ≦ defined by a ≦ b if and only if aa−1 = ab−1 is a partial order (called the natural partial order) on S and that this relation is closely related to the global structure of S (cf. (1, §7.1), (10)). Our purpose here is to study a partial order on regular semigroups that coincides with the relation defined above on inverse semigroups. It is found that this relation has properties very similar to the properties of the natural partial order on inverse semigroups. However, this relation is not, in general, compatible with the multiplication in the semigroup. We show that this is true if and only if the semigroup is pseudo-inverse (cf. (8)). We also show how this relation may be used to obtain a simple description of the finest primitive congruence and the finest completely simple congruence on a regular semigroup.

We introduce a preorder on an inverse semigroup S associated to any normal inverse subsemigroup N, that lies between the natural partial order and Green’s $${\mathcal {J}}$$ –relation. The corresponding equivalence relation $$\simeq _N$$ is not necessarily a congruence on S, but the quotient set does inherit a natural ordered groupoid structure. We show that this construction permits the factorisation of any inverse semigroup homomorphism into a composition of a quotient map and a star-injective functor, and that this decomposition implies a classification of congruences on S. We give an application to the congruence and certain normal inverse subsemigroups associate to an inverse monoid presentation.

On a Class of Lattice Ordered Inverse Semigroups

Type B semigroups are generalizations of inverse semigroups, and every inverse semigroup admits an [Formula: see text]-unitary cover (M. Petrich, Inverse Semigroups (Wiley, New York, 1984)). Motivated by studying [Formula: see text]-unitary cover for inverse semigroups, and as a continuation of Petrich’s works in inverse semigroups, in this paper, we first introduce the concept of ∗-prehomomorphism of a type B semigroup. After obtaining some basic properties, we get some structure theorems and give some conditions for a type B semigroup which is constructed by using the ∗-prehomomorphism to be proper. In particular, we introduce the notion of [Formula: see text]-unitary good cover for an abundant semigroup, and prove that every type B semigroup with compatible natural partial order admits an [Formula: see text]-unitary good cover.

Let V be any vector space and P(V) the set of all partial linear transformations defined on V, that is, all linear α: A → B, where A, B are subspaces of V. Then P(V) is a semigroup under composition, which is partially ordered by ⊆ (that is, α ⊆ β if and only if dom α ⊆ dom β and α = β | dom α). We compare this order with the so-called 'natural partial order' ≤ on P(V) and we determine their meet and join. We also describe all elements of P(V) that are minimal (or maximal) with respect to each of these four orders, and we characterize all elements that are 'compatible' with them. In addition, we answer similar questions for the semigroup T(V) consisting of all α ∈ P(V) whose domain equals V. Other orders have been defined by Petrich on any regular semigroup: three of them form a chain below ≤, and we show that two of these are equal on the semigroup P(V) and on the ring T(V). We also consider questions for these orders that are similar to those already mentioned

Given a nonempty set \(X\) and \(Y\subseteq X\), we denote by \(T(X,Y)\) the subsemigroup of the full transformation semigroup \(\mathcal{T}_X\) on \(X\) consisting of all transformations whose range is contained in \(Y\). In this paper, we endow the semigroup \(T(X,Y)\) with the natural partial order and investigate when two elements are related, then find elements which are compatible with the order. Also, we characterize the minimal elements and the maximal elements.

In a previous paper ([14]) the author showed that a free inverse semigroup is determined by its lattice of inverse subsemigroups, in the sense that for any inverse semigroup T, implies . (In fact, the lattice isomorphism is induced by an isomorphism of upon T.) In this paper the results leading up to that theorem are generalized (from completely semisimple to arbitrary inverse semigroups) and applied to various classes, including simple, fundamental and E-unitary inverse semigroups. In particular it is shown that the free product of two groups in the category of inverse semigroups is determined by its lattice of inverse subsemigroups.

GENERALIZED INVERSE SEMIGROUPS AND AMALGAMATION

We show that the variety of ortholattices has the strong amalgamation property and that the variety of orthomodular lattices has the strong Boolean amalgamation property, i.e. that two orthomodular lattices can be strongly amalgamated over a common Boolean subalgebra. We give examples to show that the variety orthomodular lattices does not have the amalgamation property and that the variety of modular ortholattices does not even have the Boolean amalgamation property. We further show that no non-Boolean variety of orthomodular lattices which is generated by orthomodular lattices of bounded height can have the Boolean amalgamation property.

SynopsisThe aim of this paper is to describe the free product of a pair G, H of groups in the category of inverse semigroups. Since any inverse semigroup generated by G and H is a homomorphic image of this semigroup, this paper can be regarded as asking how large a subcategory, of the category of inverse semigroups, is the category of groups? In this light, we show that every countable inverse semigroup is a homomorphic image of an inverse subsemigroup of the free product of two copies of the infinite cyclic group. A similar result can be obtained for arbitrary cardinalities. Hence, the category of inverse semigroups is generated, using algebraic constructions by the subcategory of groups.The main part of the paper is concerned with obtaining the structure of the free product G inv H, of two groups G, H in the category of inverse semigroups. It is shown in section 1 that G inv H is E-unitary; thus G inv H can be described in terms of its maximum group homomorphic image G gp H, the free product of G and H in the category of groups, and its semilattice of idempotents. The second section considers some properties of the semilattice of idempotents while the third applies these to obtain a representation of G inv H which is faithful except when one group is a non-trivial finite group and the other is trivial. This representation is used in section 4 to give a structure theorem for G inv H. In this section, too, the result described in the first paragraph is proved. The last section, section 5, consists of examples.

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 modified version of Bennett’s work on the structure of Schützenberger graphs of the ℛ-classes of S 1* U S 2 to state sufficient conditions for the amalgamated free products S 1* U S 2 having decidable word problem.

The absolute order on the symmetric group, constructible partially ordered sets and Cohen–Macaulay complexes

This paper presents a proof that Boolean algebras have the strong amalgamation property which does not use any choice principle, based on a feature of their binary coproducts which is derived by means of a construction from pointfree topology.

This work presents efficient algorithms to update knowledge bases in the presence of integrity constraints. The algorithms ensure that the changes to the knowledge bases are minimal. We use the deductive database paradigm to represent knowledge. Minimality is defined as a natural partial order over possible models of the database and expresses a preference for data explicity stored in the database over the data deduced by default. This requirement seems rational for many applications and yet it is hard to be expressed with a set of integrity constraints. We handle safe negation and local variables in the bodies of the rules. We tackle the issue of modifying the active domain (i.e. adding new constants or removing some of the constants which are present in the knowledge base), and we demonstrate how interaction with the user can be used to handle recursive definitions.

Several modules M over algebras with straightening law A have a structure which is similar to the structure of A itself: M has a system of generators endowed with a natural partial order, a standard basis over the ring B of coefficients, and the multiplication A × M → A satisfies a “straightening law”. We call them modules with straightening law, briefly MSLs. In section 1 we recall the notion of an algebra with straightening law together with those examples which will be important in the sequel. Section 2 contains the basic results on MSLs, whereas section 3 is devoted to examples: (i) powers of certain ideals and residue class rings with respect to them, (ii) “generic” modules defined by generic, alternating or symmetric matrices of indeterminates, (iii) certain modules related to differentials and derivations of determinantal rings. The essential homological invariant of a module is its depth. We discuss how to compute the depth of an MSL in section 4. The main tool are filtrations related to the MSL structure. The last section contains a natural strengthening of the MSL axioms which under certain circumstances leads to a straightening law on the symmetric algebra. The main examples of such modules are the “generic” modules defined by generic and alternating matrices. The notion of an MSL was introduced by the author in [Br.3] and discussed extensively during the workshop. The main differences of this survey to [Br.3] are the more detailed study of examples and the treatment of the depth of MSLs which is almost entirely missing in [Br.3]