Lower Semilattice Research Articles

Computably enumerable equivalence relations via primitive recursive reductions

Abstract The complexity classification of computably enumerable equivalence relations (or ceers, for short) has received much attention in the recent literature. A measure of complexity is typically provided by an appropriate notion of a reduction. Given binary relations $R$ and $S$ on natural numbers, a total function $f$ is a reduction from $R$ to $S$ if for arbitrary $x$ and $y$, the conditions $x~R~y$ and $f(x)~S~f(y)$ are always equivalent. If the function $f$ can be chosen primitive recursive, then we say that $R$ is primitively recursively reducible to $S$, denoted by $R \leq _{pr} S$. We investigate the degree structure $(\textbf {Ceers},\leq _{pr})$ of $\leq _{pr}$-degrees of ceers. We examine when pairs of incomparable degrees have an infimum and a supremum. In particular, we show that $(\textbf {Ceers},\leq _{pr})$ is neither an upper semilattice nor a lower semilattice. We also study first-order definable subclasses of $(\textbf {Ceers},\leq _{pr})$. In particular, we prove that the set of equivalences that have only finitely many classes is definable in $(\textbf {Ceers},\leq _{pr})$. Finally, we show that the structure of $\leq _{pr}$-degrees of computably enumerable preorders has a hereditarily undecidable theory.

On distributions of countable models for constant expansions of the dense meet-tree theory. I

We study all possible constant expansions of the structure of the dense meet-tree ⟨М; <, П⟩ [3]. Here, a dense meet-tree is a lower semilattice without the least and greatest elements. An example of this structure with the constant expansion is a theory that has exactly three pairwise non-isomorphic countable models [6], which is a good example in the context of Ehrenfeucht theories. We study all possible constant expansions of the structure of the dense meet-tree by using a general theory of classification of countable models of complete theories [7], as well as the description of the specificity for the theory of a dense-meet tree, namely, some distributions of countable models of these theories in terms of Rudin– Keisler preorders and distribution functions of numbers of limit models. In this paper, we give a new proof of the theorem that the dense meet-tree theory is countable categorical and complete, which was originally proved by Peretyat’kin. Also, this theory admits quantifier elimination since complete types are forced by a set of quantifier-free formulas, and this leads to the fact that it is decidable

INITIAL SEGMENTS OF THE DEGREES OF CEERS

AbstractIt is known that every non-universal self-full degree in the structure of the degrees of computably enumerable equivalence relations (ceers) under computable reducibility has exactly one strong minimal cover. This leaves little room for embedding wide partial orders as initial segments using self-full degrees. We show that considerably more can be done by staying entirely inside the collection of non-self-full degrees. We show that the poset can be embedded as an initial segment of the degrees of ceers with infinitely many classes. A further refinement of the proof shows that one can also embed the free distributive lattice generated by the lower semilattice as an initial segment of the degrees of ceers with infinitely many classes.

Fuzzy BCK-Algebras

In handing information regarding various aspects of uncertainty, non-classical-mathematics (fuzzy mathematics or great extension and development of classical mathematics) is considered to be a more powerful technique than classical mathematics. The non-classical mathematics, therefore, has now days become a useful tool in applications mathematics and computer science. The purpose of this paper is to apply the concept of the fuzzy sets to some algebraic structures such as an ideal, upper semilattice, lower semilattice, lattice and sub-algebra and gives some properties of these algebraic structures by using the concept of fuzzy sets. Finally, related properties are investigated in fuzzy BCK-algebras.

FINITE BCK-ALGEBRAS ARE SOLVABLE

Abstract. We present a short and very elementary proof that each finiteBCK-algebra is solvable. It is a positive answer to the question posed in[4]. Various types of BCK-algebras – as algebras strongly connected with non-classical propositional calculi – are studied by many authors. A short surveyof basic results on BCK-algebras one can find in the book [2]. The list of someunsolved (rather hard) problems on BCK-algebras one can find in [1].Let Xbe a non-empty set with a binary operation ·denoted by juxtapositionand distinguished element 0. To save the simplicity of formulaes the part ofbrackets will be replaced by dots. For example, instead of ((xy)(xz))(yz) =((xy)z)ywe’ll write (xy·xz)·yz= (xy·z)x.An algebra(X,·,0)is calledaBCK-algebra ifforanyx,y,z∈Xthe followingaxioms:(1) (xy·xz)·zy= 0,(2) (x·xy)y= 0,(3) xx= 0,(4) 0x= 0,(5) xy= yx= 0 −→x= yare satisfied.One can prove (cf. for example [2]) that in any BCK-algebra X we have(6) xy·z= xz·y,(7) x0 = x,(8) xy≤x,where ≤is a partial order on X defined by x≤y←→xy= 0.It is not difficult to see that x∧y= y·yxis a lowerbound of xand y. A BCK-algebra which is a lower semilattice with respect to ≤is called commutative.This name is motivated by the fact that such BCK-algebra is characterized bythe identity x∧y= y∧x(cf. [2] and [5]) or by the identity [x,y] = 0, where[x,y] = (x∧y)(y∧x). Properties of the operation [ , ] are described in [3] and[4].

EQ-LOGICS WITH DELTA CONNECTIVE

In this paper we continue development of formal theory of a special class offuzzy logics, called EQ-logics. Unlike fuzzy logics being extensions of theMTL-logic in which the basic connective is implication, the basic connective inEQ-logics is equivalence. Therefore, a new algebra of truth values calledEQ-algebra was developed. This is a lower semilattice with top element endowed with two binaryoperations of fuzzy equality and multiplication. EQ-algebra generalizesresiduated lattices, namely, every residuated lattice is an EQ-algebra but notvice-versa.In this paper, we introduce additional connective $logdelta$ in EQ-logics(analogous to Baaz delta connective in MTL-algebra based fuzzy logics) anddemonstrate that the resulting logic has again reasonable properties includingcompleteness. Introducing $Delta$ in EQ-logic makes it possible to prove alsogeneralized deduction theorem which otherwise does not hold in EQ-logics weakerthan MTL-logic.

UNIVERSAL COMPUTABLY ENUMERABLE EQUIVALENCE RELATIONS

Abstract We study computably enumerable equivalence relations (ceers), under the reducibility $R \le S$ if there exists a computable function f such that $x\,R\,y$ if and only if $f\left( x \right)\,\,S\,f\left( y \right)$ , for every $x,y$ . We show that the degrees of ceers under the equivalence relation generated by $\le$ form a bounded poset that is neither a lower semilattice, nor an upper semilattice, and its first-order theory is undecidable. We then study the universal ceers. We show that 1) the uniformly effectively inseparable ceers are universal, but there are effectively inseparable ceers that are not universal; 2) a ceer R is universal if and only if $R\prime \le R$ , where $R\prime$ denotes the halting jump operator introduced by Gao and Gerdes (answering an open question of Gao and Gerdes); and 3) both the index set of the universal ceers and the index set of the uniformly effectively inseparable ceers are ${\rm{\Sigma }}_3^0$ -complete (the former answering an open question of Gao and Gerdes).

Lattice operations on Rickart ∗-rings under the star order

Various authors have investigated properties of the star order (introduced by Drazin in 1978) on algebras of matrices and of bounded linear operators on a Hilbert space. Rickart involution rings (∗-rings) are a certain algebraic analogue of von Neumann algebras, which cover these particular algebras. In 1983, Janowitz proved, in particular, that, in a star-ordered Rickart ∗-ring, every pair of elements bounded from above has a meet and also a join. However, the latter conclusion seems to be based on some wrong assumption. We show that the conclusion is nevertheless correct, and provide equational descriptions of joins and meets for this case. We also present various general properties of the star order in Rickart∗-rings, give several necessary and sufficient conditions (again, equational) for a pair of elements to have a least upper bound of a special kind, and discuss the question when a star-ordered Rickart ∗-ring is a lower semilattice.

Undecidability of the elementary theory of the semilattice of GLP-words

The Lindenbaum algebra of Peano PA can be enriched by the -consistency operators which assign, to a given formula, the statement that the formula is compatible with the theory PA extended by the set of all true -sentences. In the Lindenbaum algebra of PA, a lower semilattice is generated from by the -consistency operators. We prove the undecidability of the elementary theory of this semilattice and the decidability of the elementary theory of the subsemilattice (of this semilattice) generated by the 0-consistency and -consistency operators only.Bibliography: 16 titles.

Adjoints and Monads Related to Compact Lattices and Compact Lawson Idempotent Semimodules

For compact Hausdorff Lawson lattices L that are idempotent semirings with lower semicontinuous multiplications, free compact Hausdorff Lawson L-idempotent semimodules are constructed over compacta and over compact Hausdorff Lawson lower semilattices. It is shown that these free objects are spaces of L-valued regular measures. Related left adjoint functors and monads are described.

Special classes of positive implication algebras

We construct a positive implication algebra PI( A) inherited from a chain. For a non-empty subset ∇ of a positive implication algebra, using the special set ∇( a, b), we give a necessary and sufficient condition for ∇ to be an implicative filter. We also introduce the notion of a ∇-type positive implication algebra, and prove the followings: (1) every ∇-type positive implication algebra is a commutative monoid under the operation ⊙, but not a group; (2) every ∇-type positive implication algebra is a lower semilattice; (3) in a ∇-type positive implication algebra the class of implicative filters coincides with the class of convex subsemigroups with V. Finally we show that every ∇-type positive implication algebra is a semi-Brouwerian algebra, and consider the direct product of implicative algebras.

Periodic points of nonexpansive maps and nonlinear generalizations of the Perron-Frobenius theory

Let \( K^n = \{x \in {\Bbb R}^n \mid x_i \ge 0,\ 1 \le i \le n \} \) and suppose that f : K n →K n is nonexpansive with respect to the l1-norm, \( \|x\|_1 = \sum_{i=1}^n |x_i| \), and satisfies f (0) = 0. Let P3(n) denote the (finite) set of positive integers p such that there exists f as above and a periodic point \( \xi \in K^n \) of f of minimal period p. For each n≥ 1 we use the concept of 'admissible arrays on n symbols' to define a set of positive integers Q(n) which is determined solely by number theoretical and combinatorial constraints and whose computation reduces to a finite problem. In a separate paper the sets Q(n) have been explicitly determined for 1 ≤n≤ 50, and we provide this information in an appendix. In our main theorem (Theorem 3.1) we prove that P3(n) = Q(n) for all n≥ 1. We also prove that the set Q(n) and the concept of admissible arrays are intimately connected to the set of periodic points of other classes of nonlinear maps, in particular to periodic points of maps g : Dg→Dg, where \( D{_g} \subset {\Bbb R}^n \) is a lattice (or lower semilattice) and g is a lattice (or lower semilattice) homomorphism.

On transitivity of strict preference relations

Abstract A general notion of a strict preference relation associated with a preference relation valued in a semilattice with difference is introduced. If a preference relation is valued in a linear scale then the associated strict preference relation is always transitive. In contrast, for some nonlinear scales transitivity of a preference relation does not imply transitivity of the associated strict preference relation. The identity (x ∧ y) ⊖ z =(x ⊖ z) ∧ (y ⊖ z) is characteristic for the class of lower semilattices with difference for which associated strict preference relations preserve transitivity of preference relations.

Recursion theory in a lower semilattice

In §3 we construct a universal, ℵ0-categorical recursively presented partial order with greatest lower bound operator. This gives us the unique structure which embeds every countable lower semilattice. In §§5 and 6 we investigate the recursive and recursively enumerable substructures of this structure, in particular finding a suitable definition for the simple-maximal hierarchy and giving an example of an infinite recursively enumerable substructure which does not contain any infinite recursive substructure.The idea of looking at the lattice of recursively enumerable substructures of some recursive algebraic structure was introduced by Metakides and Nerode in [5], and since then many different kinds of algebraic structures have been studied in this way, including vector spaces, Boolean algebras, groups, algebraically closed fields, and equivalence relations. Since different algebraic structures have different recursion theoretic properties, one natural question is whether an algebraic structure with relatively little structure (such as a partial order or an equivalence relation) exhibits behavior more like classical recursion theory than one with more structure (such as vector spaces or algebraically closed fields).In [6] and [7], Metakides and Remmel studied recursion theory on orderings, and, as they point out in [6], orderings differ from most other algebraic structures in that the algebraic closure operation on orderings is trivial; but this does not present a problem for them, given the questions they explore. Moreover, they take an approach of proving general theorems which can then be applied to specific orderings. Our tack is different, although also well-established (see, for example, [3]), in which a “largest” structure is defined (in §3) which corresponds to the natural numbers in classical recursion theory. In order to distinguish substructures from subsets, a function symbol is added, namely greatest lower bound. The greatest lower bound function is fundamental to the study of orderings and occurs naturally in many of them, and thus is an appropriate addition to the theory of orderings. In §4 we redefine the concepts of simple and maximal in a manner appropriate to this structure, and prove several existence theorems.

Bands with isomorphic endomorphism semigroups

Let End S denote the endomorphism semigroup of a semigroup S. If S and T are bands (i.e., idempotent semigroups), then every isomorphism of End S onto End T is induced by a uniquely determined bijection of S onto T. This bijection either preserves both the natural order relation and D -quasi-order relation on S or reverses both these relations. If S is a normal band, then the bijection induces isomorphisms of all rectangular components of S onto those of T, or it induces anti-isomorphisms of the rectangular components. If both S and T are normal bands, then S and T are isomorphic, or anti-isomorphic, or S ≅ Y × C and T ≅ Y ∗ × C , where Y is a chain considered as a lower semilattice, Y ∗ is a dual chain, and C is a rectangular band, or S ≅ Y × C, T ≅ Y ∗ × C opp , where Y, Y ∗, and C are as above and C opp is antiisomorphic to C.

Decidable subspaces and recursively enumerable subspaces

AbstractA subspace V of an infinite dimensional fully effective vector space V∞ is called decidable if V is r.e. and there exists an r.e. W such that V ⊕ W = V∞. These subspaces of V∞ are natural analogues of recursive subsets of ω. The set of r.e. subspaces forms a lattice L(V∞) and the set of decidable subspaces forms a lower semilattice S(V∞). We analyse S(V∞) and its relationship with L(V∞). We show:Proposition. Let U, V, W ∈ L(V∞) where U is infinite dimensional andU ⊕ V = W. Then there exists a decidable subspace D such that U ⊕ D = W.Corollary. Any r.e. subspace can be expressed as the direct sum of two decidable subspaces.These results allow us to show:Proposition. The first order theory of the lower semilattice of decidable subspaces, Th(S(V∞), is undecidable.This contrasts sharply with the result for recursive sets.Finally we examine various generalizations of our results. In particular we analyse S*(V∞), that is, S(V∞) modulo finite dimensional subspaces. We show S*(V∞) is not a lattice.

On the *-order for Rickart *-rings

It is shown that a Rickart *-ring forms a pseudo upper semilattice under the *-order and that a Baer *-ring in addition forms a complete lower semilattice.

On the existence of optimal fixpoints

The concept of optimal fixpoint was introduced by Manna and Shamir [6, 7] in order to extract the maximum amount of “useful” information from a recursive definition. In this paper, we extend the concept of optimal fixpoint to arbitrary posets and investigate conditions which guarantee their existence. We prove that if a poset is chain-complete and has bounded joins, then every monotonic function has an optimal fixpoint. We also provide a sort of converse which generalizes a Theorem of A. Davis [2]. If a lower semilattice has bounded joins and every monotonic function has a fixpoint, then it is chain-complete.

Tensor product of commutative unions of groups

1. Structure theorem. The study of tensor products of commutative semigroups was initiated independently by Grillet [5] and Head [6]. In 1967, T. Head [8] determined the tensor product of a commutative group with a commutative union of groups, and posed the problem of determining the tensor product of semilattices, and more generally, the tensor product of commutative unions of groups. Anderson [1] treated closely related questions. J. Delaney [4] determined the tensor product of semilattices when one of the semilattices is a chain. The solution for arbitrary semilattices (given below as Theorem 1.2) is due to N. Kimura [9], and the solution for complete lattices is forthcoming in N. Kimura and J. L. Williams [10]. In this paper, all groups and semigroups will be commutative, all semilattices will be lower semilattices with juxtaposition as the operation. Let p(A, B) denote the set of finitely generated bifilters [2] of A x B. The bifilter generated by (a, ,B) E A x B will be denoted by [(a, /3)] or by a, /8 when used as a superscript or subscript. Let T r S denote the restriction of T to S. For general information on semigroups see [3], for bifilters see [2], and for colimits see [11]. DEFINITION 1.1. Let A and B be semilattices and C = A x B be their direct product. A subset F of C is called a bifilter if the following conditions are satisfied: (KO) F is dual hereditary, i.e. if (a, b) E F, a < x, b < y then (x, y) E F. (K1) For every b E B, if (a, b), (a', b) E F then (aa', b) E F. (K2) For every a E A, if (a, b), (a, b') E F then (a, bb') E F. The following theorem is due to N. Kimura [9] and is given here without proof:

Relative annihilators in semilattices

An α-distributive (respectively α-implicative) semilattice S is a lower semilattice (with greatest lower bound denoted by juxtaposition) in which the annihilator 〈x, a〉, that is {y ∈ S: xy ≤ α}, is an ideal (respectively a principal ideal) for the fixed element α and any x of S. These semilattices appear as natural links between general and distributive semi-lattices on the one hand, and between pseudo-complemented and implicative semilattices on the other hand. Prime and dense elements, as well as maximal and prime filters, are essential. Mandelker's result, a lattice L is distributive if and only if 〈x, y〉 is an ideal for any x, y ∈ L is extended to semi-lattices.