### On the calculation of evolutionarily stable strategies.

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Bounded lattices with an antitone involution the complemented elements of which do not form a sublattice must contain two complemented elements such that not both their join and their meet are complemented. We distinguish (up to symmetry) eight cases and in each of these cases we present such a lattice of minimal cardinality.

We show that every idempotent weakly divisible residuated lattice satisfying the double negation law can be transformed into an orthomodular lattice. The converse holds if adjointness is replaced by conditional adjointness. Moreover, we show that every positive right residuated lattice satisfying the double negation law and two further simple identities can be converted into an orthomodular lattice. In this case, also the converse statement is true and the corresponence is nearly one-to-one.

Certain ring-like structures, so-called orthorings, are introduced which are in a natural one-to-one correspondence with lattices with 0 every principal ideal of which is an ortholattice. This correspondence generalizes the well-known bijection between Boolean rings and Boolean algebras. It turns out that orthorings have nice congruence and ideal properties.

It is shown that in the variety of orthomodular lattices every hypersubstitution respecting all absorption laws either leaves the lattice operations unchanged or interchanges join and meet. Further, in a variety of lattices with an involutory antiautomorphism a semigroup generated by three involutory hypersubstitutions is described.

Brouwerian semilattices are meet-semilattices with 1 in which every element a has a relative pseudocomplement with respect to every element b, i. e. a greatest element c with a∧c ≤ b. Properties of classes of reflexive and compatible binary relations, especially of congruences of such algebras are described and an abstract characterization of congruence classes via ideals is obtained.

Ring-like operations are introduced in pseudocomplemented semilattices in such a way that in the case of Boolean pseudocomplemented semilattices one obtains the corresponding Boolean ring operations. Properties of these ring-like operations are derived and a characterization of Boolean pseudocomplemented semilattices in terms of these operations is given. Finally, ideals in the ring-like structures are defined and characterized.

A variety is called normal if no laws of the form $s=t$ are valid in it where $s$ is a variable and $t$ is not a variable. Let $L$ denote the lattice of all varieties of monounary algebras $(A,f)$ and let $V$ be a non-trivial non-normal element of $L$. Then $V$ is of the form $\mathrm{M}od({f}^{n}\left(x\right)=x)$ with some $n>0$. It is shown that the smallest normal variety containing $V$ is contained in $\mathrm{H}SC\left(\mathrm{M}od({f}^{mn}\left(x\right)=x)\right)$ for every $m>1$ where $\mathrm{C}$ denotes the operator of forming choice algebras. Moreover, it is proved that the sublattice of $L$ consisting of all normal elements of...

It is proved that orthomodular posets are in a natural one-to-one correspondence with certain residuated structures.

Relational systems containing one binary relation are investigated. Quotient relational systems are introduced and some of their properties are characterized. Moreover, homomorphisms, strong mappings and cone preserving mappings are introduced and the interplay between these notions is considered. Finally, the connection between directed relational systems and corresponding groupoids is investigated.

States on commutative basic algebras were considered in the literature as generalizations of states on MV-algebras. It was a natural question if states exist also on basic algebras which are not commutative. We answer this question in the positive and give several examples of such basic algebras and their states. We prove elementary properties of states on basic algebras. Moreover, we introduce the concept of a state-morphism and characterize it among states. For basic algebras which are the certain...

By a relational system we mean a couple $(A,R)$ where $A$ is a set and $R$ is a binary relation on $A$, i.e. $R\subseteq A\times A$. To every directed relational system $\mathcal{A}=(A,R)$ we assign a groupoid $\mathcal{G}\left(\mathcal{A}\right)=(A,\xb7)$ on the same base set where $xy=y$ if and only if $(x,y)\in R$. We characterize basic properties of $R$ by means of identities satisfied by $\mathcal{G}\left(\mathcal{A}\right)$ and show how homomorphisms between those groupoids are related to certain homomorphisms of relational systems.

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