An Application of le-semigroup Techniques to Semigroups, Γ-semigroups and to Hypersemigroups

An $le$-semigroup, is a semigroup $S$ at the same time a lattice with a greatest element $e$ ($e\ge a$ for every $a\in S$) such that $a(b\vee c)=ab\vee ac$ and $(a\vee b)c=ac\vee bc$ for all $a,b,c\in S$. If $S$ is not a lattice but only an upper semilattice ($\vee$-semilattice), then is called $\vee e$-semigroup. A $poe$-semigroup is a semigroup $S$ at the same time an ordered set with a greatest element $e$ such that $a\le b$ implies $ac\le bc$ and $ca\le cb$ for all $c\in S$. Every $\vee e$-semigroup is a $poe$-semigroup. If $S$ is a semigroup or a $\Gamma$-semigroup, then the set ${\cal P}(S)$ of all subsets of $S$ is an $le$-semigroup. If $S$ is an hypersemigroup, then the set ${\cal P^*}(S)$ of all nonempty subsets of $S$ is an $le$-semigroup. So all the results of $le$-semigroups, $\vee e$-semigroups and $poe$-semigroups based on ideal elements, automatically hold for semigroups, $\Gamma$-semigroups and hypersemigroups. This is not the case for ordered $\Gamma$-semigroups or ordered hypersemigroups; however the main idea, even in these cases, comes from the $le$ ($\vee e$)-semigroups. As an example, we study the weakly prime ideal elements of a $\vee e$-semigroup and their role to the different type of semigroups mentioned above.

Comparing Voting by Committees According to Their Manipulability

We consider the class of voting by committees to be used by a society to collectively choose a subset from a given set of objects. We offer a simple criterion to compare two voting by committees without dummy agents according to their manipulability. This criterion is based on the set-inclusion relationships between the two corresponding pairs of sets of objects, those at which each agent is decisive and those at which each agent is vetoer. We show that the binary relation “to be as manipulable as” endows the set of equivalence classes of anonymous voting by committees (i.e., voting by quotas) with a complete upper semilattice structure, whose supremum is the equivalence class containing all voting by quotas with the property that the quota of each object is strictly larger than one and strictly lower than the number of agents. Finally, we extend the comparability criterion to the full class of all voting by committees. (JEL D71, D72)

DEFINABILITY OF THE JUMP OPERATOR IN THE ENUMERATION DEGREES

We show that the e-degree 0'e and the map u ↦ u' are definable in the upper semilattice of all e-degrees. The class of total e-degrees ≥0'e is also definable.

Embedding jump upper semilattices into the Turing degrees

We prove that every countable jump upper semilattice can be embedded in , where a jump upper semilattice (jusl) is an upper semilattice endowed with a strictly increasing and monotone unary operator that we call jump, and is the jusl of Turing degrees. As a corollary we get that the existential theory of 〈D, ≤T, ∨, ′〉 is decidable. We also prove that this result is not true about jusls with 0, by proving that not every quantifier free 1-type of jusl with 0 is realized in . On the other hand, we show that every quantifier free 1-type of jump partial ordering (jpo) with 0 is realized in . Moreover, we show that if every quantifier free type, p(x1,…, xn), of jpo with 0, which contains the formula x1 ≤ 0(m) & … & xn ≤ 0(m) for some m, is realized in , then every quantifier free type of jpo with 0 is realized in .We also study the question of whether every jusl with the c.p.p. and size is embeddable in . We show that for the answer is no, and that for κ = ℵ1 it is independent of ZFC. (It is true if MA(κ) holds.)

Non-distributive upper semilattice of Kleene degrees

K denotes the upper semilattice of all Kleene degrees. Under ZF + AD + DC. K is well-ordered and deg(XSJ) is the next Kleene degree above deg(X) for X ⊆ ωω (see [4] and [5, Chapter V]). While, without AD, properties of K are not always clear. In this note, we prove the non-distributivity of K under ZFC (§1), and that of Kleene degrees between deg(X) and deg(XSJ) for some X under ZFC + CH (§2.3).